Theresa May has accused the European Union of not treating the UK with respect in a deliberately combative address that prompted a sharp fall in the pound amid fears that it made a no deal Brexit more likely.

Twenty-four hours after her Salzburg humiliation, the prime minister gave a hastily arranged televised Downing Street statement in an effort to reassert herself.

Going on the offensive, she blamed EU leaders for the “impasse” in negotiations, a phrase that spooked the currency markets and led sterling to fall 1.5% against the dollar, its biggest one day drop this year.

Speaking from a lectern with two union flags behind her, May tried to accuse her counterparts of bad faith: “I have treated the EU with nothing but respect. The UK expects the same. A good relationship at the end of this process depends on it.”

Her remarks came a day after the Salzburg summit where EU leaders frustrated by May’s insistence on sticking rigidly to her controversial Chequers negotiating plan embarrassed her when the EU council president, Donald Tusk, unexpectedly declared it “would not work”.

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May retaliated on Friday: “Yesterday, Donald Tusk said our proposals would undermine the single market. He didn’t explain how in any detail or make any counter-proposal. So we are at an impasse.” She called on the EU to spell out “what their alternative is”.

But Tusk, the European council president, hit back hard at the prime minister, accusing her of bringing a “surprisingly tough and in fact uncompromising” attitude to the Salzburg summit. He denied May’s claims that the EU had failed to explain its position to the British negotiators, claiming to have spelled out the problems in detail in recent weeks, although he insisted that “a compromise, good for all, is still possible”.

Despite the heated rhetoric and plunge in sterling, EU officials are understood to be working on a counter-proposal to the Chequers plan, which is likely to appear in early October, after the Conservative party conference and a week before a make or break Brexit summit in Brussels.

No 10 is calculating May can win support by appearing to stand up to the EU, although her critics in the party believe this would be the moment to give up the Chequers plan, or risk facing hostile activists at the party’s annual conference in just over a week.

Several cabinet ministers put out statements of support. Penny Mourdant, the international development secretary, again emphasised the possibility of a no deal, saying that while voters “still want a deal” they would be “content to go without one, even folk who voted remain”.

But Jacob Rees-Mogg, the chair of the European Reform Group of hard Brexit Tories, said that Salzburg was a failure and repeated that it was time to abandon Chequers. “There is still no reason to suppose that Chequers can work either for the UK or the EU.”

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Chequers proposes that the UK shares a common rulebook for goods and food after Brexit in an attempt to prevent a return of customs checks for goods crossing the Irish border, a plan so unpopular on the right of the Tory party that Boris Johnson and David Davis resigned rather than support it.

The Brussels objections to Chequers centre on the fact that the common rule book only covers a fraction of the regulations covering goods – in the chemical sector for instance the figure is estimated at 30%.

That would imply the UK would have to align itself more broadly to EU rules after Brexit to avoid a return to a hard border in Ireland, a proposal that could be difficult for May to sell to her party.

Hard Brexiters will step up their campaign against Chequers in the run up to the Tory party conference on Monday. D Davis and other senior rightwing politicians are expected to attend an event at which alternative “Canada plus” trade proposals will be announced.

The EU, however, has said that while it would be prepared to discuss a Canada-style deal, that would mean not just taking the UK out of the single market and customs union, it would also mean the return of border controls in Ireland.

The prime minister also signalled that the UK would unilaterally safeguard the rights of the EU citizens living in the country in an attempt to reassure them that the impasse in negotiations would not affect their status.

“There are over 3 million EU citizens living in the UK who will be understandably worried about what the outcome of yesterday’s summit means for their future. I want to be clear with you that even in the event of no deal your rights will be protected. You are our friends, our neighbours, our colleagues. We want you to stay,” May said.

Ministers are expected to confirm details in a couple of weeks, after the party conference season.

The prime minister repeatedly rejected an EU backstop proposal that would see Northern Ireland remain inside the customs union if the UK was unable to agree a free trade agreement with Brussels in the divorce talks.

Ratcheting up the language on the issue, May said: “It is something I will never agree to – indeed, in my judgment it is something no British prime minister would ever agree to. If the EU believe I will, they are making a fundamental mistake.”

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EU diplomats warned that the prime minister’s statement only made a no-deal scenario more likely, and expressed their astonishment at May’s bellicose tone. “She is now playing into a negative dynamic,” one diplomat said. “My best assessment is that rhetoric won’t bring an orderly exit. I think she should have taken that away from the 27 approach [in Salzburg].”

In a rare direct criticism of a serving prime minister, Tusk made reference to the criticisms of Michel Barnier by the British government, and the refusal of the British side to work on a solution to the Irish border issue by October, as previously promised.

“The UK stance presented just before and during the Salzburg meeting was surprisingly tough and in fact uncompromising”, he said. “The response of the EU27 leaders was to reiterate our trust in chief negotiator Michel Barnier and to reiterate our position on the integrity of the single market and the Irish backstop.

But he added: “While understanding the logic of the negotiations, I remain convinced that a compromise, good for all, is still possible. I say these words as a close friend of the UK and a true admirer of PM May.”

Jeremy Corbyn, the Labour leader, criticised both the prime minister and the EU: “Theresa May’s Brexit negotiating strategy has been a disaster. The Tories have spent more time arguing among themselves than negotiating with the EU.

“From day one, the prime minister has looked incapable of delivering a good Brexit deal for Britain. The political games from both the EU and our government need to end because no deal is not an option.”