In British-English slang:
– the phrase to disappear (also to vanish, etc.) up one’s own arse means: to become self-involved, pretentious or conceited;
– the phrase to be (stuck) up one’s own arse means: to be self-involved, pretentious or conceited.

For example, the Scottish chef Gordon Ramsay (born 1966) used the phrase to be (stuck) up one’s own arse in an interview about his television series, Next Level Chef (featuring the English chef Paul Ainsworth (born 1979)), which was to be broadcast on ITV—interview by David Ellis, published in the Evening Standard (London, England) of Wednesday 11^th January 2023 [page 26, column 5]:

A humble Ramsay seems a rare thing, but the series appears to promise it, especially as Ainsworth gives his former mentor hell. Will we see the Scotsman get his comeuppance? Ramsay laughs: “I’ve never been so far up my own arse that I’ve not wanted to be challenged.” Who comes out on top? Ah, of course. Should have guessed—we’ll have to watch to find out.

The earliest occurrences that I have found of the phrases to disappear (also to vanish, etc.) up one’s own arse and to be (stuck) up one’s own arse are as follows, in chronological order:

1-: From Excluded from the Cemetery (London: Hutchinson & Co., Ltd., 1966), by the British novelist Peter Marshall (1939-1972) [Part Two: Joe; Chapter 3, page 181]—however, it seems to me that, here, the phrase to disappear up one’s own arse means to totally disappear:

If my memories are false, memories on which my whole life depends, if they are false, I say, how can I kill myself? I don’t want to kill a stranger, do I? Not before I get to know him, anyway. I’m curious about this stranger living in my skin. That’s logical, isn’t it? Isn’t it? (It’s a funny thing, logic, said Mad Joe, seriously. If you apply logic to it, that is, if you push it to its logical extremes, you will invariably find a paradox. This stranger, should you ever meet him, might not be a stranger, after all. You see, logic cannot stand the application of logic. Under such an application, logic will disappear up its own arse—)

2-: From Private Parts and Public Places (London: Hutchinson & Co., 1967), by the British author Robin Cook (1931-1994)—as reprinted in 1972 by Panther Books Limited, London [chapter 1, page 19]:

Millionaires, he thought, bored him almost as much as film stars; they were obsessed with themselves to such a degree that it was no wonder they ended by vanishing up their own arse-hole.

3-: From the column Londoner’s Diary, published in the Evening Standard (London, England) of Wednesday 4^th October 1989 [page 6, column 6]:

Branagh ¹ as Rambo ²
A SPECIAL meet-the-actors lunch following the screening of Branagh’s Henry V yesterday was mildly post-nuclear as actors and guests stayed away. “Kenny” Branagh did turn up to give his views on the movie, however.
“I want this film to appeal to all the Rambo fanatics out there,” he insisted, an interpretation overlooked by Shakespearean scholars until now. Branagh’s best chum Brian Blessed threw in his ha’penny worth. “The trouble with theatre is that it has a tendency to disappear up its own arse,” observed Blessed.
“The trouble with Shakespeare is that it’s ruined by schools and the priests of art who regard it as some sort of religion. This film is going to change all that”.

¹ Kenneth Branagh (born 1960) is a Northern-Irish actor and film director.
² John Rambo is a fictitious U.S.-Army veteran interpreted by the U.S. actor Sylvester Stallone (born 1946) in several U.S. action films.

4-: From Frank, earnest and rude, by Deborah Orr, published in The Guardian (London and Manchester, England) of Tuesday 27^th August 1991 [page 27, column 1]:

The announcement of Skinner ³ as this year’s Perrier Award winner has been greeted in many quarters with the same outrage as would be the annointing [sic] of Gennady Yanayev ⁴ with a history of theatre prize. One could be forgiven for thinking it’s a sign the Perrier Award for outstanding Fringe comedy/revue started 11 years ago, just at the time when “alternative comedy” is generally agreed to have emerged. Now, at the comic high point of a festival in which comedy has come under heavy fire, we’re back at the start with a dirty gag merchant. Could it be that the alternative scene is, as Frank might ever so dangerously say, disappearing up its own arse? Apparently not.
“Frank Skinner has taken the traditional northern working-men’s club act,” enthused one of the 10 Perrier judges, “and subtly reinvented the genre as something fresh, right on.”

³ Frank Skinner (Christopher Graham Collins – born 1957) is a British comedian.
⁴ The Russian politician Gennady Yanayev (1937-2010) was the disputed President of the Union of Soviet Socialist Republics for three days (from 19^th to 21^st August 1991).

5-: From Radio, by Val Arnold-Forster, published in The Guardian (London and Manchester, England) of Friday 29^th November 1991 [page 38, column 8]:

Radio 1 had a very sparky debate on Monday: Is Pop Art? It included all sorts of persons from the world of pop, a clutch of young listeners and a man from the Arts Council who, not surprisingly, collected a lot of flak. I can’t say I heard many new arguments. We even had that old cost-of-opera chestnut reroasted. But Nicky Campbell was a skilful manoeuvrer of his wayward charges, even when they saw themselves (possibly correctly) as “in great danger of disappearing up our own arses”.

6-: From an account of the “Always There” gig held by Que Pasa at a warehouse in east London, by Marianne Macdonald, published in The Independent (London, England) of Monday 8^th February 1993 [No. 1,967, page 2, column 5]:

“I don’t usually like London crowds. They’re too up their own arses,” said Breeze, one of the 10 DJs paid to play until morning.

7-: From Taking the riff raff with the smooth: Snob’s law sets up a class act, a review of the British television comedy-drama The Riff Raff Element, broadcast on BBC1—review by Alec Lom, published in the Daily Mirror (London, England) of Saturday 27^th March 1993 [TV Weekly; page 2, column 6]:

The Belchers, an unruly mob from a sleazy council estate, make a stark contrast to the upper-crust Tundish family.
They pick pockets and burgle houses on nearby posh estates to earn an “honest” living.
[…]
To them the distinctly upper middle-class Tundishes are “off their heads and up their own arses”.