Jeremy Hunt has told leaders of the European Union to not “mistake British politeness for weakness” after the tense Salzburg summit that threatens to sink Theresa May’s Chequers proposal.

Speaking on BBC Radio 4’s Today programme, the foreign secretary followed May’s combative address on Friday, saying: “What Theresa May is saying is: don’t mistake British politeness for weakness. If you put us in a difficult corner, we will stand our ground. That’s the kind of country we are.”

Hunt said the government understood the EU’s red lines, but that did not mean the UK was not able to agree to a trading relationship where trade was in effect frictionless. “We’ve made a democratic decision and this government, Theresa May, is not going to sign up to a deal that is not consistent with the spirit and the letter of the referendum,” he said.

He called on EU leaders to get the tone right in the negotiations and avoid “insulting [May], insulting the British people on social media. Getting into these standoffs where you’re calling people liars and so on.”

When asked if he was referring to Donald Tusk’s Instagram post, Hunt said yes. “A piece of cake, perhaps?” Tusk asked in the post, alongside a picture of him and May at the summit in Austria. “Sorry, no cherries.”

Hunt said: “We need to avoid revving up the situation, making it worse by appealing to audiences on social media.” Instead, he said, all parties should be “seriously and diligently working to a solution”.

“That’s what Theresa May wants to do and she has taken some big risks. She’s shown real flexibility with these Chequers proposals, she lost two important cabinet ministers as a result of that, and we need the EU to show flexibility in return. If there are elements of our proposal they find difficult, then tell us what they are and we will sit down and work through them.”

In a hastily arranged address on Friday, May accused the EU of not treating the UK with respect and blamed them for an impasse in negotiations. “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,” she said. Her address prompted a sharp fall in the pound, increasing fears that a no-deal Brexit is now more likely.

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While the UK is keen to avoid such a scenario, Hunt said: “Let’s be clear, even in a situation where we aren’t able to come to an agreement, we would be trading on World Trade Organization terms and it would be bumpy, it would be difficult, but we would find a way to survive and prosper as a country. We’ve had far bigger challenges in our history.”

Earlier on the programme, the former foreign secretary David Miliband called on Labour to be “much stronger” on Brexit and backed a second referendum. “The truth is that waiting for the government to foul up is no strategy at all. The country desperately needs a strong government, but it also needs a strong opposition,” he said.

Describing a no-deal scenario as a “terrible disaster for the country”, he added that it was “an absolute dereliction of duty, in my view, for the Labour party leadership not to embrace the fundamental principle that, since the Brexit the people were sold two years ago is not available, it’s essential the Brexit deal the prime minister does is put to people.”