Theresa May has tried to threaten EU leaders over dinner at a special summit in Salzburg by telling them the UK would not seek to delay Brexit, prompting European leaders to warn that the two sides remained far apart on trade and the Irish border despite months of negotiations.

The prime minister told her counterparts “that the UK will leave on 29 March next year” and as a result “the onus is now on all of us to get this deal done” by the end of an emergency summit that the EU confirmed would happen in mid-November.

It was the first time since Chequers that May has had a chance to address the EU’s other 27 heads of government instead of going through their chief negotiator, Michel Barnier, with No 10 hoping that it would inject some urgency into the divorce talks. “We all recognise that time is short but extending or delaying these negotiations is not an option,” she said.

But as the summit started Jean-Claude Juncker, the president of the European commission, said that a deal remained “far away” while Donald Tusk, the president of the European council, warned that the UK’s proposals for the Irish border and future trade relations with the EU needed to be “reworked and further negotiated”. Tusk added that “various scenarios are still possible” – a clear hint that no deal was still a possibility.

Despite the EU leaders’ statements, No 10 is hoping that May’s pitch to EU leaders will eventually prompt some greater flexibility on the part of Brussels in the critical period for the Brexit negotiations between a scheduled European council meeting in October and the decisive summit in November.

Theresa May would then have to get her deal signed off by parliament – the so-called “meaningful final vote” – before the government can then implement the legislation it needs in time to hit the deadline of leaving the EU on 29 March 2019, two years after serving notice under article 50.

The summit also saw May reject in front of her peers the proposals put forward by Barnier for a revised Irish backstop border because the EU is still insisting on customs checks in the Irish Sea if the two sides cannot strike a free trade agreement after Brexit.

“The idea that I should assent to the legal separation of the United Kingdom into two customs territories is not credible,” May said following a dinner of schnitzel and fruit salad in the Austrian city. A long debate over migration left only a small avenue for the prime minister to give her views on the Brexit talks. A source said that she talked for just five to ten minutes.

There was no debate – EU leaders will instead return to the subject of Brexit tomorrow lunchtime where they will discuss the issue without the British leader present.

May was also able to refer directly to her Chequers proposals for the UK’s future relationship with the EU, in which the UK would sign up to “a common rulebook” for food and goods after Brexit, in the hope that other leaders would engage with them more positively than has happened until now.

While there is a consensus among the EU27 that the Chequers’ central proposals for a common rulebook and a complicated customs arrangement do not offer a way forward, the countries with the closest trade ties to the UK have called for the EU to bridge the gap in positions rather than merely insisting on a Canada-style free trade deal.

Countries including Sweden, the Netherlands, Belgium and Denmark argue that the UK has “evolved” its position. Tusk reflected that stance on Wednesday when he said Chequers indicated a “positive evolution” in British thinking.

But hours before she arrived, May was reminded how unpopular the plans are with her party’s rightwing when the former Brexit secretary, David Davis, released remarks from a speech he will give in Munich on Thursday, in which he says the prime minister’s Chequers plan is unpopular and does not represent what people voted for at the time of the referendum in 2016.

Davis says the prime minister had previously promised to “return control over our law, our money and our borders. These promises were in [the Conservative] manifesto too. But the Chequers plan crosses on all of those red lines. The EU is often correctly described as having a democratic deficit. But Chequers is devoid of democracy altogether.”

The former minister, who resigned when May promised to impose Chequers, will say in a speech to a German thinktank that “many of us will shortly be presenting an alternative plan which will outline a more ambitious vision” and said that the UK and Europe would be better off if they engaged in friendly economic competition.

Complications over implementing Brexit were further highlighted by junior Treasury minister Mel Stride, who diverted from the prime minister’s position by appearing to suggest a second referendum could happen if parliament voted down the deal May negotiates with Brussels.

Stride said: “When we have a firm deal on the table, I suspect that those to the right of the party – the pro-Brexit wing – will be very concerned that if that deal does not prevail, they will end up in the situation where we could have a second referendum or we could end up not leaving the EU altogether.”

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Angela Merkel, the German chancellor, also offered little succour to Downing Street over progress on the future trading relationship with the EU. She said: “I hope that we have an exit that takes place in a good atmosphere, with great respect for each other and that in certain areas very close cooperation is possible, namely in the areas of security – of domestic and also foreign security.”

In Salzburg, the prime minister will hold bilateral meetings on Thursday with the Irish taoiseach, Leo Varadkar, over breakfast, and with Tusk after the lunch from which she is excluded as she seeks to end the impasse.

On Tuesday, Barnier said he was making revised proposals as part of an attempt to “de-dramatise” the Irish border issue, and tried to downplay them by describing them as “a set of technical checks and controls”, insisting that the EU respected the territorial integrity of the UK.

According to EU sources, Barnier briefed ministers from the 27 member states on his fresh thinking on the Irish backstop at a meeting on Tuesday evening. The EU’s chief negotiator said that while there were four categories of products that would be under scrutiny when coming into Northern Ireland from the rest of the UK, only animal and food products would always require physical checks at the border.

Customs declarations, VAT declarations and tests on the conformity of industrial goods to EU standards could be made in advance. There would only be a need for random physical tests at ports based on a risk analysis, Barnier told EU ministers. One senior EU diplomat said that this was the last word on the issue from their side, and that they would continue to push it, even though May has repeatedly said it was unacceptable.