A senior Downing Street source said the Commons’ support would have “helped the PM in her arguments in Brussels”.

“It’s important for MPs to vote for it so we can give the right signal to Brussels,” the source said ahead of the vote.

Instead, the vote exposed deep divisions within May’s Conservative party, and shredded her authority as she seeks to guide Brexit through to its March 29 deadline.

The original withdrawal agreement she struck with the EU last year was rejected by parliament in January. May then voted against her own deal, winning the backing of parliament to return to the EU and seek changes to the ‘backstop’ designed to keep the Irish border free of checkpoints.

The government believes that if the backstop is given a time limit, or a unilateral withdrawal clause, or is replaced with a technological solution, then it could get the divorce deal through parliament. v

But until now the EU has insisted it would not reopen the withdrawal agreement that it struck with the UK government last November.

This was May’s tenth loss in the Commons. She lost by 258 to 303, deserted by both Brexit hardliners and pro-Remain MPs from her own Conservative party in the Thursday evening vote.

She was not in the Commons for the vote, having been advised of the likelihood of a loss.

A Downing Street spokesman put the blame on Labour, saying by voting against the government’s motion they were “in effect voting to make no deal more likely”.

However Corbyn said May needed to “tell us what the plan actually is”.

“I was going to ask her to come to the despatch box now and admit that her strategy has failed,” Corbyn said. “The government cannot go on ignoring parliament or ploughing on without a coherent plan.”

Labour MP Chuka Umunna said an extension to the Brexit March 29 deadline was now “imperative”, but it was hard to see how any Brexit plan would get a majority in parliament without another referendum.

The vote was preceded by a rancorous debate.

Brexit secretary Stephen Barclay had told parliament to “hold its nerve”.

“We need to send a clear signal to those in the European Union with whom we are discussing these issues,” he said.

Brexiters were angry that the motion put forward by the PM for the vote endorsed the view that there must not be a ‘no deal’ Brexit.

Several MPs said this effectively tied the PM’s hands behind her back in negotiations.

But Barclay disagreed, saying a no-deal Brexit would still happen by force of law if the parliament did not vote for a withdrawal deal.

Shadow Brexit secretary Kier Starmer said there was a “casualness” about discussing the prospect of no deal.

“[It is an attitude] that we can somehow, in a macho way, march off the cliff and it will all be fine, it will be so good for the country,” he said.

“We have to take action if we are serious about avoiding the calamity and catastrophe of no deal… it is obvious what the prime minister is up to. She is pretending to make progress while running down the clock.

“The simple and painful truth is this: if there had been a viable alternative to the backstop, there would never have been a backstop.”

Conservative MP Ken Clarke said the idea of “going for the catastrophe of no deal on the arbitrary date of 29 March… is ridiculous”.

But the parliament’s attempts to make progress had turned into an appalling shambles, he said.

“We have not yet identified, and certainly not demonstrated, a clear majority for any particular course,” Clarke said. “We are not being given many opportunities to do so, and we keep retreating when we get given them.

“Nobody here is able to agree in sufficient numbers on what on earth they want to do. All we are doing is vetoing each other’s propositions on what should go forward.”