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LONDON (CNN) — There was no sign of a breakthrough in the Brexit deadlock Wednesday when U.K. lawmakers failed to agree on an alternative to Theresa May's beleaguered plan even as she offered to resign in a desperate attempt to save it.

Facing intense pressure from her fractured party, the British Prime Minister said she would step down if and when her Brexit deal was delivered. There was a desire for "a new approach, and new leadership," she told a meeting of lawmakers from her Conservative Party.

Hours later, members of the House of Commons voted on a range of measures designed to break the impasse over Brexit — but failed to agree on any of them.

The day ended in chaos as MPs bickered over whether to hold another day of debate under an unprecedented process outside the control of the government.

Resignation offer

Downing Street hoped that May's offer to resign would persuade enough rebel Conservative MPs to change their minds and back her deal, which was rejected by a margin of 149 votes when it was last put before Parliament.

In the immediate aftermath of her announcement, key lawmakers fell into line. Boris Johnson, the colorful former Foreign Secretary, who quit over her handling of Brexit, said he would reluctantly support May's deal. Others, including former party leader Iain Duncan-Smith, followed his lead.

Downing Street plans to put May's Withdrawal Agreement back before Parliament before the end of the week. But its prospects were dealt a heavy blow when the Democratic Unionist Party, the hardline Northern Irish group that props up May's minority government, announced it was still implacably opposed to it.

The DUP is fundamentally opposed to a key provision in the deal — the so-called Irish backstop, which would keep Northern Ireland in a customs union with the E.U. in the event of a failure to agree the terms of a future relationship with the U.K. The "necessary changes ... to the backstop have not been secured," and the risk of being "trapped in the backstop" was high, it said.

Without the DUP's support, May will find it hard to pass her deal. Under an agreement struck with the E.U. to extend the Brexit process, it must be ratified by Friday.

'Desire for a new approach'

May's big gamble — the big surprise of the day — came at a meeting of the 1922 Committee, the influential group of all backbench Conservative Members of Parliament. Her decision to resign would enable lawmakers to complete their "historic duty" and "deliver on the decision of the British people and leave the European Union with a smooth and orderly exit," she said.

"I am prepared to leave this job earlier than I intended in order to do what is right for our country and our party," May told them. "I know there is a desire for a new approach — and new leadership — in the second phase of the Brexit negotiations — and I won't stand in the way of that."

Conservative Party lawmaker Simon Hart told CNN that the atmosphere in the room was "respectful" and May "was as animated as I'd ever seen her at committee. She was passionate but not emotional. We are all traumatized at this point by these negotiations."

But with the DUP's decision to oppose her deal, it was unclear whether her resignation would be enough. And given her offer to quit was contingent on the House of Commons passing her deal, May faced the unusual situation of being deeply unpopular with the party, yet not being able to muster enough votes to ensure her resignation.

Even if May could garner enough support to pass her deal, it faces procedural hurdles before it can be put to a third vote. Speaker of the House John Bercow has ruled that it must be substantively different from the previous two occasions to qualify for a repeat showing. Bercow repeated the ruling on Wednesday.

Brexit Secretary Steven Barclay implied earlier that MPs will vote on May's deal for a third time on Friday, saying that a motion to enable debate for that day would be put before the House of Commons on Thursday. (Friday is usually a day off.)

Conservative lawmaker Oliver Letwin, the architect of the plan to seize control of Wednesday's parliamentary business and hold a series of "indicative" votes, said it was a matter of "great disappointment" that none of them could command a majority. But amid angry scenes in the House of Commons, he insisted that a plan to hold a second day of debate, on Monday, would go ahead.

The only plan that came close to succeeding was a proposal that Britain would remain within a permanent customs union with the E.U. after Brexit. It was rejected by a eight votes — 272 votes to 264.

A plan to hold a confirmatory referendum on any agreed Brexit deal lost by 27 votes — 268 for, and 295 against. Backers of a second referendum insisted that it proved their plan was gaining support.