It's just one long list of things to get done in Anguilla

There's Johnno's on Friday, Pumphouse on Saturday, The Dune on Sunday. Is there any rest for visitors to this Caribbean island? Katy Guest packed it all in

The first thing that a visitor must do on arriving in Anguilla is to find an extra day in the week. Everyone on the island – from the driver who picks you up from the airport to the drunk fella at the rum shack – will add at least one more essential item to your itinerary.

It's the Pumphouse on Saturdays for rum & gingers, says the cool New Yorker who shows you round the hotel. It's Johnno's on Friday for ribs and garlic bread, insist locals at the beach. It's Seliki for lobster, Barrel Stay for puddings, and, according to a Yorkshire couple who are honorary locals, Roy's at Sandy Ground for the best fish and chips outside God's Own County. It's The Dune on Sundays where "almost anything goes", most say. And "don't plan anything else on Sunday", they all add, knowingly.

If all this sounds a little bit stressful, don't worry: Anguilla is not a place for making a list and ticking things off it. Here, stuff just seems to happen. The local airline, Liat, which transfers Brits and Americans from nearby Antigua, is fondly known around here as Leave Island Any Time. But in my experience it left the island much as it arrived, disappointingly punctually. On Anguilla, everything is so smooth, air conditioned and friendly that it's as if the whole island is moving on invisible castors, with a strict brief to make something cool and unexpected appear just around every corner.

Only one thing seems to put the locals off their stride and that's when you tell them that you're staying at the Viceroy, Anguilla's latest hotel to open. Then they say "wow!" or "you jammy git" or make a sort of whistling sound through their teeth. Glistening away on the south-west corner of the skinny, 16-mile island, the brand new Viceroy really is something else. Its long, palm-lined driveway has a 10mph speed limit, as if taking the place in too quickly might cause heart attacks. From reception, the view stretches the other way, across plump green lawns and a geometric pool to the sea.

Apparently, the rod-straight palm trees grew within braces until recently. Based on squareness, solidity, clean, straight lines and epic vistas, the Viceroy is designed impeccably by Kelly Wearstler. The interiors reflect its seafront location, as does the glass-sided restaurant (it's all about the kobe beef), with its tall and flimsy walls that barely seem to keep out the elements. Driftwood lamps and sea-glass mirrors soften the vast, light rooms. Every view leads through something towards the sea.

Unfortunately, my favourite villa (private mini beach; double showers both in and outdoors; back bedroom for your staff ...) has just been sold for $10m and rents for $10,000 (£6,900) a night during the high season. But regular rooms and suites are also suitably impressive, many with their own private plunge pools. Critics would say that the Viceroy could be anywhere: Dubai, Las Vegas, Singapore. But frankly, when you're floating in a rooftop plunge pool and drinking champagne, who really cares which country you're in?

It is tempting, of course, to slide into that pool and never get out. But get out you must – because beyond the waterfall at the Viceroy's gate is a whole different world of meals to be eaten and rum to be drunk.

At this point we have to address Anguilla's main failing: its public transport. The failing is that there isn't any. At least, on the main road that runs up and down the spine of the island, I saw one bus stop; but at no point did I see a bus. Taxis apparently exist, but at a minimum $40 for a run back from the pub, they're prohibitively expensive. Walking anywhere on the roads at night would be scary, especially when you're told: "Oh, everyone on Anguilla just drives drunk."

And you can add to this the three dogs who roam the highway and whose main job is to lie on the tarmac and scare the rum punch out of unwary drunkies. A hire car with a designated driver, clearly, is the way to go.

Most of the nightlife on Anguilla happens on Sandy Ground – a thin strip of exactly that, bordered on one side by a shallow salt pond and on the other by the Caribbean Sea. In most countries whose main industry is tourism, there are places where the tourists go to drink, and places where the locals go to get away from them. On Sandy Ground, the two groups genuinely seem to mix. It's the kind of place where tourists come back year after year – for example, Sparkie and David, who hang out at the swish Veya restaurant and at "the best jerk chicken stall" opposite the People's Market in The Valley, and who seem to know all the gossip there is to know on Anguilla. They say that there are places to stay here more modest than the Viceroy – such as the colourful Lloyds, starting at $99 per night (that's 267 eastern Caribbean dollars, or ECD), and the gloriously English-colonial Cap Juluca on the south coast.

If you're going to cram in every place that's recommended, then, you're going to have to work up an appetite. Start at The Pumphouse, a small, dark wooden building with a stage for live music, a maritime vibe and a dizzying array of rums. Ask for Pyrat, the local tipple, to seem in the know. From there, stagger to Barrel Stay to dine in front of the beach, the sea and a straight-from-the-brochure Caribbean sunset.

Is it the rum talking, or could this be the best restaurant in the world? The British owners, Jill and Graham, had never heard of Anguilla before he went to view the storm-hit beachside canteen for sale, five years ago, and phoned her to say: "This is it."

It took Jill three weeks to sell up and move out. Graham had trained at the Savoy company and was the head chef at L'Escargot in Soho; his skill seems to be in turning the freshest of fish and the most prestigious of ingredients (Valrhona chocolate; Scottish beef; Italian truffles) into dishes so extravagantly perfect that you want to weep. Jill's skill is in reciting the specials menu, from memory, with such lascivious enthusiasm that it's hard not to order everything. Several diners have asked to record her.

If you can still stand after seafood ceviche, blackened fillet of Mahi Mahi and a "rich dark home-made chocolate tart with mild chilli and lime sauce", for which I would gladly sell my soul, the evening ends at Elvis's. You'll find it at the end of a handwritten sign marked Lonely Street – just next door to the Heartbreak Hotel – and a strict notice reading: "Respect! No parking here, mon". The band plays West Indian music, as well as British and American classics that sound West Indian enough as the sun goes down, the rums get bigger, and a beatific dog heaves himself up from the sand dancefloor and plonks himself under the wooden bar. It's about now that you're glad you remembered that designated driver.

The next morning might not seem the best time for an early rise and a day's sailing. Especially when you arrive to find the dog from last night, sleeping off his hangover under a pier. But Captain Rollins Ruan, the skipper of the 35ft catamaran Chocolat, will soon snap you out of it. He's a tolerant man, up to a point – he drops anchor to let a couple off for a sojourn on a deserted cay; prepares lunch and cold beers on deck, and even allows a desperately keen Englishman to take the wheel of his precious boat (and is grudgingly complimentary about his sailing). But don't get Captain Ruan started on yacht one-upmanship. "That," he will spit, pointing out a huge and hideous shiny monstrosity rumoured to be owned by an upstart young Russian, "is a ugly ass thing."

The day after such exertion must be taken slowly. Over goat curry at Smokey's, right on one of the island's huge, near-deserted beaches, a local explained that they make a sort of herbal Viagra out of the sea moss here, and added: "We have fared much better as an island of quality rather than quantity. The government in the 1980s, at the start of the tourism boom, decided: no casinos, no cruise tourism, no large commercial shopping, no hawkers touting wares. And we feel good about that." That seemed truer than the thing about the sea moss.

After lunch, a bright-green shed at the side of the road in the capital, The Valley, caught our attention. It turned out to be a rum shack called Moving Violation Top-Up (Mind, Body & Soul), owned by a President Obama fan named Victor who keeps herbs and tomatoes, growing upside down in plastic squash bottles hanging from his porch. On weekdays, they serve jerk chicken, salt fish, rice and peas, and diners play backgammon on the rickety tables. How much for two large rums, we ask. "Where are you staying?" "The Viceroy." "That'll be 10 dollars." Doh!

In a suitably surreal state of mind, we head to The Dune. Bankie Banx, the most famous musician on Anguilla – when he's not busy touring the world – holds court at this rambling, homemade, treehouse-cum-shipwreck of a bar, put together out of driftwood and hurricane-wrecked boats. The house band, British Dependency, jam "No Woman No Cry", "Stand By Me" and "To Love Somebody" as we find ourselves sitting on the edge of a wooden verandah, looking out across the sea at a silhouetted St Martin, and swinging our legs over the sand. Bankie gets up from his sofa by the bar, rum in hand, and swaggers over to play a number with the band. His innumerable offspring run in and out from the dunes. A beautiful woman kisses him. Englishmen try to dance.

At some point the sun goes down behind the dune. A little later the rum runs out. Anything may or may not have gone – almost. The list remains largely unticked. But if an extra day could be found in the week, this is what it would be for.

Compact Facts

How to get there

Katy Guest travelled to Anguilla courtesy of ITC Classics (01244 355527; itcclassics.co.uk), which offers eight nights for the price of four from 5 July to 31 October, at £1,750 per person, saving £965 per person. The price is based on two adults sharing a one-bedroom Resort View Villa and includes return flights with British Airways from London Gatwick to Antigua, internal flights and private return transfers. Island Car Rentals (islandcar.ai) offers car hire from $45 per day, plus $6 insurance and $20 for a temporary Anguilla driving licence.

Further information

Sail Chocolat – Captain Rollins Ruan (001 264 497 3394; ruan@ anguillanet.com) charges US$80 (£55) per person for a day excursion to the cays, $60 for a Sunset sail, and $300 for private cruises and charters for one to four people.