Video report by ITV News Europe Editor James Mates
The UK and the EU are running out of time to strike a Brexit deal after European council president Donald Tusk warned Theresa May her plan "will not work".
Dismissing her Chequers proposal, Mr Tusk set an alarming deadline for progress, saying: "The moment of truth for Brexit negotiations will be the October European Council. In October, we expect maximum progress and results in the Brexit talks."
Mrs May says her Brexit plan is the "only serious and credible" proposal on the table and she was backed on Thursday night by Leave-backing Transport Secretary Chris Grayling, who said there are no changes “on the table at the moment”.
The Chequers plan, agreed by the Cabinet in July, involves keeping parts of the single market after Britain leaves the European Union - something Mrs May has been told cannot be cherry-picked.
In fact, Mr Tusk appeared to mock Mrs May over this notion on Instagram, posting a photo of the pair looking at cakes with the caption: “A piece of cake, perhaps? Sorry, no cherries.”
The warnings from EU leaders over Chequers are the latest blow to the negotiations and Mrs May's leadership.
Many Tory MPs have been unhappy with the plan for weeks - it prompted leading Brexiters David Davis and Boris Johnson to resign from the Cabinet - and the prime minister has been warned by allies that it is "dead as a dodo" with little chance of passing through the House of Commons.
After a meeting of EU leaders in Salzburg, Mr Tusk said of the Chequers proposals: "The suggested framework for economic cooperation will not work, not least because it risks undermining the single market."
Watch Donald Tusk's damning statement below
Mrs May remains wedded to the Chequers blueprint, noting it is the only proposal on the table as the deadline approaches, although she indicated the UK will unveil new measures on the future status of the Northern Irish border in a bid to break the deadlock.
She has flatly rejected a European Commission backstop proposal for Northern Ireland to remain within the EU customs area after Brexit, arguing this would draw a border down the Irish Sea.
She told reporters: "Our White Paper remains the only serious and credible proposition on the table for achieving that objective."
When asked if the chances of a no-deal Brexit had now increased, Mrs May remained positive but conceded: "There's a lot of work to be done."
Mr Grayling said the Government would not drop its Chequers plans in the face of “tough” language, adding the EU’s demands on Northern Ireland are “impossible” for the UK to accept.
ITV News Political Editor Robert Peston believes "the UK and EU are still million miles apart on Ireland backstop" - with month left before the deadline.
Mrs May also rejected calls from some EU leaders for the Government to call a second referendum.
She said: "There will be no second referendum. There has been a vote of the people, it took place in June 2016 and people voted to leave the European Union.
"Of course, you have heard those voices from Europe that talk about a second referendum.
"Actually, I think others have started to recognise rather more this is going to happen. We are going to leave the European Union."
Questions of a second referendum will be a further source of frustration for the Prime Minister after dismissing the idea on Wednesday.
Frustration over Brexit was also shared by French President Emmanuel Macron who attacked the Leave campaign in the 2016 referendum, saying the people who told British voters that Brexit would be easy were "liars".
In a press conference in Salzburg, Mr Macron said: "Brexit is the choice of the British people and it is a choice pushed by certain people who predicted easy solutions.
"Brexit has shown us one thing - and I fully respect British sovereignty in saying this - it has demonstrated that those who said you can easily do without Europe, that it will all go very well, that it is easy and there will be lots of money, are liars.
"This is all the more true because they left the next day, so they didn't have to manage it."
At home, there was condemnation on both sides of the political divide for Mrs May's plan.
Shadow Brexit secretary Sir Keir Starmer believes it has put the country in a precarious position: "It has been clear for weeks that Theresa May's Chequers' proposals cannot deliver the comprehensive plan we need to protect jobs, the economy and avoid a hard border in Northern Ireland.
"With just weeks to go until a deal must be struck, the Prime Minister cannot keep ignoring this reality. She needs to urgently drop her reckless red lines and put forward a credible plan for Brexit."
While Conservative Brexiteer Jacob Rees-Mogg said Mr Tusk had reinforced the objections to the Chequers plan set out by the EU's chief negotiator Michel Barnier.
"I think Chequers now has no supporters at all. I doubt even the Downing Street cat is any longer backing the Chequers plan.
"I think the time has come for Mrs May to say 'This is not going to work'."