Downing Street says she stands by her commitment to see through this stage of Britain’s exit, and some colleagues say she is going nowhere.

“I don’t think we should be rushing to change our leader when there is a big task to be done,” Justice Secretary David Gauke told the BBC ahead of the summit.

May herself told a Brussels press conference that she “sincerely regrets” the delay but “we must now press on at pace with our efforts to reach a consensus on a deal”.

But many of May’s Conservative MPs are livid that Britain is now likely to take part in European Parliament elections next month – something she previously said would be unacceptable.

Why a Brexit delay could come back to bite Europe

Brexit-supporting MP Peter Bone earlier noted May said she would not accept staying in the EU beyond June 30.

“If the prime minister intends to keep her word, can we expect her resignation later tonight?” he tweeted.

May’s move to reach a compromise with the opposition Labour Party to find votes to replace those she has lost on her own side has also provoked accusations of “surrender”.

And, Brexit anger aside, amany of May’s Conservative MPs are livid that Britain is now likely to take part in European Parliament elections next month.

May has faced constant challenges since taking office following the historic 2016 referendum vote to leave the EU, presiding over a divided party and since 2017, without a majority in parliament.

More than 30 ministers have quit since the 2017 election, with some – notably Boris Johnson – becoming the strongest critics of her European strategy.

Her failure to get her EU divorce deal through parliament in recent months has left her authority in tatters, unable to control legislation or even her own ministers.

In a vote on the eve of the EU summit, 97 Conservative MPs opposed May’s original plan to delay Brexit until June 30, and around 80 more abstained.

EU leaders rejected this date after a lengthy discussion in Brussels, but the new date of October 31 is shorter than some proposed.

‘Sofa, so bad’: watch laid-back UK PM Theresa May speak to Britain from a comfy couch as her country sinks into Brexit oblivion

This potentially scuppers any MPs hoping to use an extension to replace May with someone who would play tougher with the EU.

And there is no obvious way for May’s Conservative critics to replace her.

She survived a confidence vote within her Conservative party in December, making her immune for a year.

Labour leader Jeremy Corbyn also tried to oust the government in January but May’s MPs rallied around, unwilling to risk another election.

She has hinted she might voluntarily step down if she fails to get a Brexit deal through by June 30.

But the pressure to quit might come earlier, if the Conservatives do badly at local elections on May 2.

Aides emphasise her determination to deliver Brexit, but many believe it is May’s rigidity that is blocking any compromise deal with MPs.

Professor Meg Russell, of University College London’s Constitution Unit, noted that leading a minority government requires a “flexible, nimble approach, including the ability to compromise and to work cross-party”.

“Theresa May clearly has none of these skills,” she wrote.

However, her premiership has been marked by her dogged refusal to give up, and few commentators would rule out the prospect that she tries to stay on for as long as possible.

Who among Tories could replace May?

Boris Johnson

Still the favourite for the top job, Johnson has kept himself out of some of the messiest Tory warfare over the last few weeks and has enthusiastic support from younger Brexiter MPs – and the patronage of Jacob Rees-Mogg.

His supporters insist no other name on the list can connect with voters in the same way and win a general election.

However, his reputation is still severely damaged from his time as foreign secretary and there is a concerted “anyone but Boris ” campaign among party colleagues.

Matt Hancock

While the response of many voters to mention of the health secretary is still likely to be “Who?”, to some he is starting to have the makings of a from-the-sidelines contender.

The former culture secretary is only 40 but has six years of frontbench experience, and is on to his second cabinet post.

Amid a trio of hopefuls speaking at an event about young voters Hancock had the best lines, saying voting Conservative “used to be something people started to think about doing when they got their first pay cheque – now it’s when they get their first winter fuel allowance”.

Dominic Raab

The former Brexit secretary has a loyal fan base and a professional team, including support from Vote Leave’s ex-comms director Paul Stephenson.

MPs are forming the view that the next party leader should be a younger face from a new generation of politicians – which gives Raab the edge over Boris Johnson.

However, he is resented by some MPs for holding out too long before backing the PM’s Brexit deal.

Tom Tugendhat

It is Tugendhat’s achievement so far to be considered among the rising contenders while indicating he is not necessarily going for the top job, as he did at the event about young voters this week.

At the same event Tugendhat showed why some nonetheless think he is a possible future leader despite being a backbencher (albeit one who chairs the foreign affairs committee), answering questions thoughtfully and fluently.

Liz Truss

Truss has worked the circuit more than almost any other potential candidate, speaking on the future of the Tory party at almost any venue that will have her – think tanks, launches and in weekend newspaper spreads.

She hopes to win Brexiter backing with a publicly enthusiastic embrace of a no-deal Brexit – much to the scepticism of Treasury colleagues.

Michael Gove

The fortunes of the environment secretary remain hard to predict and opinion is split in the party. His detractors believe he is deeply unpopular with the country and has ruined his reputation for good when he stood against Johnson at the last leadership race.

However, most MPs were delighted by his performance in the no-confidence vote where he tore into Corbyn.

“That united the party for the first – and only – time in months,” one MP said.

“It made us realise what we could have.”

Additional reporting by The Guardian