Boris Johnson has warned Conservative MPs that their party faces an “existential threat” if they fail to deliver Brexit, as he vowed to take Britain out of the EU on 31 October with or without a deal.

The Tory leadership frontrunner dismissed suggestions that he could support a second Brexit referendum, saying another public poll was “absolutely anti-democratic”.

Speaking at the first hustings of the Conservative leadership contest, Mr Johnson told Tory MPs that their party was “starring down the barrel” and warned: “The death watch beetle has entered the timbers of the Tory party.”

He said the Tories must “get Brexit done and lance that boil”, claiming the Conservatives would “pay a terrible price” if they delayed Britain’s EU departure any further.

Addressing around 80 MPs at an event organised by the One Nation group of Conservatives, the former foreign secretary said that, as prime minister, he would seek to renegotiate Theresa May‘s Brexit deal, which he said “simply does not work”.

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He insisted he did not want a no-deal outcome but warned Tory MPs their party could be destroyed if they fail to deliver Brexit.

He said: “We are facing an existential crisis and will not be forgiven if we do not deliver Brexit on 31 October.

“We need to realise the depth of the problems we face – unless we get on and do this thing, we will be punished for a very long time. There is a very real choice between getting Brexit done and the potential extinction of this great party.”

He promised to deliver Brexit by 31 October, saying: “The public are fed up to the back teeth with kicking the can down the road. They’ve had enough of delay.

“I really think we will pay a terrible price if we delay. People will spot that the legitimacy of the referendum is starting to wane and pressure will grow from those who feel there should be another vote, which I completely reject.

“We’ve got to get this thing done. We’re staring down the barrel ... We’ve got to get Brexit done and lance that boil.”

Mr Johnson explicitly ruled out both an early general election and another Brexit referendum.

Asked whether he could back a Final Say vote if it is the only way to secure a deal, he said: “I fought the first campaign. It was quite a tough experience and a rough experience for the country. A lot of us have been a bit bloodied and it’s been very divisive. I don’t want us to go through that again. To have a second referendum is absolutely anti-democratic.”

Instead, he insisted he would try to reopen negotiations with the EU, claiming there was new pressure from key member states to agree a deal that the UK parliament could support.

He said: “You can already see in France and Germany there is much greater pressure [to find a solution].”

If the EU rejected attempts to renegotiate, he said, it would be “all the more reason to declare for no-deal”, insisting the UK needed “to be committed, to show resolve”.

Speaking afterwards, Gavin Williamson, the former defence secretary who is now leading efforts to lobby MPs on Mr Johnson’s behalf, said: “What all Conservatives recognise is that we’ve got to deliver Brexit and it’s becoming increasingly clear to colleagues that he’s the only one who can deliver it, he’s the only one that can sell the message, he’s the only one that can put the Brexit Party back in its box and he’s the only one who can win the next general election and keep Corbyn out of Downing Street.

“You see a dawning realisation among so many colleagues as you go around the corridors that Boris is the only one who is going to be able to save the party.”

Nicky Morgan, the former education secretary, one of the leaders of the One Nation group, said Mr Johnson had been “subdued”. She said: “He was serious. Sometimes there is a Boris who is larger than life. I think he took the seriousness of the situation and the hustings and responded quite rightly.

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Earlier in the hustings, Sajid Javid, the home secretary, also ruled out a general election and said he too wanted to renegotiate Ms May’s Brexit deal to add a time limit or exit clause to the controversial Northern Ireland backstop.

He said he would also focus on creating more opportunities for people in deprived areas outside London, including coastal towns and typically white working class areas.

He said: “We can’t be a one nation party if we’ve got two economies – London and the southeast, and then everywhere else. When I talk about unifying the country, I’m talking about giving everybody an opportunity to get on – not just giving people a chance to make their way in the capital.”