In her article, she writes of taking part in the “haute route”, a 111-mile ski trek from Chamonix in France to Zermatt in Switzerland. Earlier this year she took part in a similar, shorter, event called the Engadin, a ski marathon in St Moritz.

“Back home in London after my Alpine challenges, I can now pursue less demanding hobbies in my spare time, such as ping-pong,” she writes.

“I’m informed that Boris Johnson, former editor of this magazine, wants to be ‘whiff-whaff’ world king even more than he wants to be prime minister.

“I’m also told the Johnsons are almost as competitive as the Middletons. So I’d like to lay down a challenge to the Mayor.

“My only stipulation is that I can use my favourite Dunlop Blackstorm Nemesis bat, which I used when I played in the Milton Keynes U13 National Championships, don’t you know. Bring it on, Boris.”

The challenge is the first time Miss Middleton has spoken of her enjoyment of ping-pong, although she and her sister, the Duchess of Cambridge, are known to be sporty, having played hockey at school.

However, Miss Middleton, 29, has become better known throwing herself enthusiastically into her celebrity lifestyle, and last month started writing a column in Waitrose Food Illustrated magazine, after her book on entertaining, which became the subject of widespread ridicule.

In her Spectator diary she mocked her first recipes for Waitrose, which included sushi and desserts made of sake, as she described her diet during the five-day ski challenge.

“For at least a week before any major physical challenge, I try to load up on carbs to boost my energy levels,” she said.

“This time, I ditched the usual diet of sushi, Vietnamese spring rolls and tangerine and saké jellies, and feasted on Alpine food, which I love: tartiflette, rosti and raclette. Nothing better.

“Yet what really sustained me on the Engadin and the Haute Route were the energy gel packs — tubes of unpleasant fruity slime. It’s like drinking mayonnaise in a Frubes packet. I have been to Snow+Rock several times this year already just to stock up on this horrid substance. The thought of it now gives me shivers, but it keeps you going.”

The mention of a retail outlet will raise questions over whether she is acting as a celebrity “endorser” for the shop.

In her diary she hints at the level of interest she experiences from public relations companies, joking that her rucksack was weighed down during the ski trip by “my Cheltenham fur hat, various other bits of clothing that PR people have sent me, and several signed copies of my book Celebrate”.

The diary also makes no mention of her new boyfriend, Nico Jackson, the 35-year-old son of a stockbroker, instead noting the presence of “our strikingly handsome guide, Matthieu”.

She also confesses that she failed to finish, due to bad weather and the threat of avalanches, after five nights spent in “dormitories with numbered beds; women alongside men, who are generally unwashed and snoring”, adding: “Another year, perhaps.”