"This is the only deal possible", says European Commission President Jean-Claude Juncker.
A proposal for a new customs union got close to passing in a first round on Wednesday, as did a plan for a referendum on May's deal, with more voting planned next Monday and Wednesday.
Later in the evening there was a small stand-off between Robinson supporters and police, as around 100 supporters protested outside May's office shouting "We want Brexit!".
Labour leader Jeremy Corbyn has called on Mrs May to change her deal or resign immediately, while Northern Ireland's DUP - which has propped up Mrs May's minority government - also continues to oppose the deal.
However, a No 10 source indicated the prime minister would continue to seek support in the Commons.
The 2016 referendum vote to leave the European Union revealed a United Kingdom divided over many more issues, and has provoked impassioned debate about everything from secession and immigration to capitalism, empire and what it means to be British. The tricky thing though will be getting a fourth vote past the Commons Speaker, who already tried to block the third vote and is not keen for MPs to vote on the same motion again.
If a majority emerges for a softer Brexit, as expected, May will have to decide whether to do as she is told and risk a rebellion; or call an election on the basis that Parliament is asking her to deliver something that goes against previous campaign pledges.
Every parliamentary defeat on her deal since it was signed with Brussels four months ago, and the ongoing stalemate in the Commons, is due to that disastrous decision to hold an election and its hung parliament result.
May referred to that process after the defeat on Friday.
But maybe soon it will be her leadership, her deal, that has passed its limits. Should she follow through on her televised address following the second meaningful vote defeat and ask the electorate to instruct its MPs to back the Brext deal - a referendum between remaining in the European Union or leaving on the basis of the negotiated deal - or should she accept that her government simply can not command majority support in the Commons for the defining policy of her premiership and trigger the process for an early general election?
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Mrs May said Friday's vote means Britons will "almost certainly" have to elect MEPs to the European Parliament in elections taking place at the end of May.
The BBC said government officials had not ruled out the possibility of a run-off vote in parliament between the most popular option proposed by lawmakers and May's deal. Parliament on Monday will vote on various alternatives to May's strategy.
Even Nigel Farage, a British driving force behind Brexit and staunch European Union opponent, has nothing but admiration for Barnier who kept 27 nations aligned as one while Britain, as one, crumbled into chaos.
Downing Street said Mrs May would continue to talk to the Democratic Unionist Party about more reassurances over the backstop - the "insurance policy" created to prevent physical infrastructure at the Irish border.
But 34 Tories still rebelled, and Northern Ireland's DUP stood firm against it because of concerns about the Irish border backstop.
Sterling had been mostly stronger against both the dollar and the euro in the runup to the vote, but then fell to its lowest point against the greenback in more than two weeks within moments of the count.
"The prime minister has run out of the road".
"We are talking with them about trade and we can do a very big trade deal with the United Kingdom", he said. May said there would be "grave" implications.
They included some of her staunchest critics such as former foreign minister Boris Johnson - a leading contender to replace her - and MP Jacob Rees-Mogg.