Theresa May has said it is still possible to get the assurances MPs need to back her Brexit deal, despite EU leaders ruling out any renegotiation.

At a summit in Brussels, the UK PM said there was “work to do” but talks on “further clarification” would continue.

She admitted a “robust” discussion with European Commission President Jean-Claude Juncker, after he complained about “nebulous and imprecise debate”.

Labour said the withdrawal deal was now “dead in the water”.

The UK prime minister travelled to Brussels to make a special plea to EU leaders after delaying Tuesday’s Commons vote on the deal, in anticipation of a heavy defeat.

She then went on to win a confidence vote brought by her own MPs but vowed to listen to the concerns of the 37% of Tory MPs who voted against her.

Many of them are concerned that the controversial “backstop” plan in the withdrawal agreement Mrs May has negotiated, which is aimed at preventing a hard border in Northern Ireland, would keep the UK tied to EU rules indefinitely and curb its ability to strike trade deals.

What was the “nebulous” row about?

After Mrs May addressed EU leaders at the summit on Thursday evening, Mr Juncker urged further clarity from the UK.

He said: “Our UK friends need to say what they want, instead of asking us to say what we want… because this debate is sometimes nebulous and imprecise.”

Video footage of the two on Friday morning captured a tense exchange, apparently about his remarks, although the exact words were not audible.

Asked about what she had said to him, Mrs May told reporters: “I had a robust discussion with Jean-Claude Juncker – I think that’s the sort of discussion you’re able to have when you have developed a working relationship and you work well together.

“And what came out of that was his clarity that actually he’d been talking – when he used that particular phrase – he’d been talking about a general level of debate.”

At a later press conference, Mr Juncker described Mrs May as a “good friend” who he admired as a “woman of courage”.

He said he hadn’t realised nebulous was a word in English and he had been referring, not to her, but to the “overall state of the debate in Britain”.

He said: “I can’t see where the British parliament is heading. That’s why I was saying that it was nebulous – foggy in English – I was not addressing her.”

He also said: “We have to bring down the temperature” of the debate amid “attacks coming from Westminster against Europe and the European Commission”.

Did the PM secure any concessions?

And a paragraph which had appeared in the draft conclusions at the start of the summit, saying that the EU “stands ready to examine whether any further assurance can be provided” did not appear in the final conclusions.

But Mrs May said that, despite reports that the EU was unwilling to consider further clarification, she had talked to Mr Tusk, Mr Juncker and others that morning which “have shown that further clarification and discussion following the council’s conclusions is in fact possible”.

She also welcomed commitments by other EU leaders to try to get a new trade deal in place “speedily” so that the backstop would not be needed and said that, as formal conclusions from the summit, they had “legal status”.

But she added: “There is work to be done. It is clear we can look at this issue of further clarification. We will be working expeditiously over the coming days to seek those further assurances I believe MPs will need.”

BBC Brussels reporter Adam Fleming said the fact that the EU said it would use its “best endeavours” to get a future trade deal that would get rid of the need for a backstop – even if the backstop came into force – was seen as important by British officials who said it meant the UK could go to an independent arbitration panel if they felt the EU was dragging its feet.

What do the deal’s critics say?

Labour leader Jeremy Corbyn said: “The last 24 hours have confirmed that Theresa May’s Brexit deal is dead in the water. The prime minister has utterly failed in her attempts to deliver any meaningful changes to her botched deal.

“Rather than ploughing ahead and dangerously running down the clock, the prime minister needs to put her deal to a vote next week so Parliament can take back control.”

And Conservative Brexiteer Mark Francois told the BBC: “It is as plain as a pikestaff that this will never get through the House of Commons… the prime minister, I’m afraid, is completely boxed in.”

The Democratic Unionist Party, on whom Theresa May relies for her Commons majority, said she must deliver “legally binding changes” to the withdrawal agreement if she wants her deal to get through Parliament.

Its leader Arlene Foster said the prime minister “has made commitments” to the DUP and “knows what she has to deliver on”.

Meanwhile, the UK and Switzerland have provisionally agreed to keep their current trading rules after Brexit, the first of 40 existing EU trade deals with other countries the UK hopes to adopt.

The agreement, which will replicate the EU’s current arrangement with Switzerland as closely as possible, is due to come into place at the end of 2020 when the Brexit transition phase ends.

But it could come into force at the end of March if the UK leaves with no deal.

Analysis: Not a failure but unlikely to impress critics

The BBC’s Brussels reporter Adam Fleming

Theresa May’s mission to Brussels has not been a failure.

She has a written statement from her 27 fellow leaders confirming – reconfirming, really – that the Irish backstop is an insurance policy which would only ever be temporary.

If the backstop is activated, then the EU would use its “best endeavours” to negotiate a trade deal, which would mean it could be deactivated. That wording is crucial, say British officials, because it means the UK could refer the EU to the independent arbitration panel established in the Brexit treaty if London felt Brussels was moving too slowly.

The EU dropped a commitment to look for further ways to help the UK, which means there won’t be a formal process to find them.

But it doesn’t mean the search couldn’t happen informally, or in private, or at the last minute.

The problem is that these commitments are unlikely to impress Theresa May’s harshest critics. And they certainly wouldn’t fit on the side of the bus as reasons to sign up to Mrs May’s Brexit deal.