Just as every East End boozer has a photograph of The Queen Mother pulling a pint from behind its bar, so every great European hotel likes to claim that Audrey Hepburn once stayed there. And just like you wonder how the real-life royal found the time to pull a pint in every pub within the sound of Bow Bells (She didn’t. She only pulled a pint in one, The Queen’s Head, a beaten-up corner-pub opposite my old gaff in Stepney), you question how the Queen of Hollywood managed to holiday at every grande dame from the Baltic coast to the Côte d’Azur. Did cinema’s pencil-slim princess not have a home to go to? Well, for more than a decade, Switzerland’s Bürgenstock Resort was Hepburn’s home.

After marrying her first husband, actor-director Mel Ferrer, at a chapel on the estate in 1954, Hepburn moved into the glitzy mountain resort. Each day, according to legend, Hepburn would drive to a local dentist – although no one seems to know which ailment would require daily visits to an orthodontist.

The next highest ranking card a hotel can draw in a game of Iconic Guest Top Trumps is Sophia Loren. Extraordinarily, the other leading lady of Hollywood’s Golden Age also lived at the Bürgenstock Resort. After marrying her own producer-husband, Carlo Ponti, in 1957, Loren moved into a wooden villa near where Hepburn got hitched. She stayed there for seven years.

We don’t know if Hepburn and Loren ever fought over sun loungers around the hotel’s lake-facing outdoor pool (given that it’s now known as the ‘Hollywood Pool’, we might assume they both were regulars). Nor do we know if the actresses ever engaged in a Génépi-fuelled dance-off at the resort’s wood-panelled Spycher nightclub (once a granary, now a private event space). It’d be nice to think that Hepburn was good enough to share some tips on good dental practise, but we don’t know that either. What we do know is that neither Hepburn nor Loren took a selfie in the spa’s wraparound infinity pool. Smartphones didn’t exist then. Neither did the infinity pool.

Today, you can take as many pictures of yourself floating in the sky as you like, but only between 5pm and 6pm. Such is the Bürgenstock’s staggering situation, straddling a ridge, crocodile-like, 500 metres above Lake Lucerne, that you could, and would, spend the entire day capturing yourself morphing into a human prune if they allowed you to do so. To avoid selfie sticks at dusk, the resort has had to enforce a strict phone curfew. And enforce it they will. I got a rollicking at five minutes past six.

If it doesn’t feel like you’re staying in a resort that dates back to 1873, that’s because most of it doesn’t. After its Sixties heyday, shiny Bürgenstock started to lose its lustre, eventually filing for bankruptcy in the Nineties. Swiss investment banks tend not to touch hospitality ventures. Too much risk. Surprisingly, UBS made a successful bid for the pile in 1996. A decade later, having worked out just how much it would cost to modernise the dilapidated, 148-acre estate, the bank offloaded the site to the Qataris (far less risk-averse, the Qataris). Yet even the owners of Canary Wharf and Heathrow Airport nearly cut their losses halfway through a regeneration project that became known in Switzerland as ‘the project of the century.’

Connecting the three-storey spa, one of the largest in the world, to the 10-storey Bürgenstock, the swankiest of the resort’s four hotels, is a glass walkway straight out of Tony Stark’s house. It’s part of a redevelopment that eventually took nine years, 148 building permits and £440 million to get over the line. It opened in August 2017 to include a nine-hole golf course, tennis courts, five pools, 70km of walking and biking trails, and one of Europe’s largest, and most exclusive, luxury medical facilities. Psychosomatic therapy, which focuses on mental health, is one of the most popular treatments.

Head sommelier Matteo Rimoldi takes care of a 114-page wine list. For some guests, he says, the wine is more of a draw than the spa. Rimoldi stocks 87 types of champagne and 108 different Bordeauxs, including an ’82 Mouton Rothschild (£2,800) and an ’89 Pétrus (£10,075). He doesn’t need to bother with Prosecco, he jokes – but means it – and keeps his scarce Swiss wines for his favourite guests. Mike Wehrle, the Bürgenstock’s high-spirited Corporate Culinary Director, heads up seven restaurants and a team of 95 chefs. Watching a company of them pirouette around the enormous open kitchen of the hotel’s principal restaurant-in-the-sky, Spices, is Swan Lake in black aprons.

Some guests come for the spa, others for the wine. I’d go back for the bath. When outfitting each room, the astute minds at London-based MKV Design noticed that the views up here are pretty top. So they plonked a tub right next to the bathroom window. To the side of ours was a double-sided glass fireplace, meaning you could look across to flames licking at a stone surround, and peer down at dinky little boats drawing lines on the lake.

You wake in the clouds and lather up in the stars.

Rooms from approx. £1,200 per night while staying in the Bürgenstock Hotel, visit burgenstockresort.com

Read more: Why the Italian Dolomites are the ideal year-round destination