Speaking at a private hustings, Johnson told party members he is the man to put Farage ‘back in his box’

Boris Johnson has warned the Conservative party it is facing extinction if it does not deliver Brexit and put Nigel Farage “back in his box” as he spoke at a hustings of candidates to be the next prime minister from which the press and public were banned.

Johnson told MPs from the centrist One Nation wing of the party that it was crucial to take the UK out of the EU at the end of October, arguing that he was the candidate to see off both Farage and Labour’s Jeremy Corbyn.

Ruling out calling an early general election or discontinuing parliament to push through a no-deal Brexit, Johnson said: “We are looking at a very difficult situation and we must get ready, eventually, but not immediately, to beat Jeremy Corbyn and put Farage back in his box. We are facing an existential crisis and will not be forgiven if we do not deliver Brexit on 31 October.

Quick guide Tory party leadership contest Show Hide What happens next in the Tory party leadership race? As she announced on 24 May, Theresa May stepped down formally as Conservative leader on Friday 7 June, although she remains in place as prime minister until her successor is chosen. In a Conservative leadership contest MPs hold a series of votes, in order to narrow down the initially crowded field to two leadership hopefuls, who go to a postal ballot of members.

How does the voting work? MPs choose one candidate, in a secret ballot held in a committee room in the House of Commons. The votes are tallied and the results announced on the same day. In the first round any candidate who won the support of less than 17 MPs was eliminated. In the second round anybody reaching less than 33 votes was eliminated. In subsequent rounds the bottom placed contender drops out until there are only two left. The party membership then chooses between them. When will the results be announced?

The postal ballot of members has begun, and the Tory party says it will announce the new prime minister on 23 July.. Photograph: Neil Hall/EPA

“I believe I am best placed to lift this party, beat Jeremy Corbyn and excite people about conservatism and Conservative values. 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. But I believe I can take on Farage and win back the voters being won over by him.”

Despite having run the leave campaign, Johnson said the Tories needed to “stop banging on about Brexit and put that bawling baby to bed, pacify it and recapture the political agenda with one-nation conservatism”.

Sajid Javid, the home secretary, made a similar pitch that the Conservatives could not “beat the Brexit party by becoming the Brexit party”.

Nicky Morgan, one of the Tory MPs who convened the One Nation hustings, said Javid, Andrea Leadsom and Rory Stewart had joined Johnson in ruling out an early general election and proroguing parliament to push through a no-deal Brexit.

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Speaking outside the meeting, Morgan defended the right of Tory MPs to have private hustings of candidates to be prime minister, saying “most job interviews are done in private”. Tory sources said media were not allowed in because some candidates had objected to the idea of the press reporting their statements.

The journalist Katy Balls, who writes for the right-leaning Spectator magazine, conducted the questioning of candidates, but no other reporters were allowed in to listen to the answers.

The One Nation group will hold a series of further private hustings with the candidates, but Johnson is shaping up to be the clear frontrunner, with the endorsement of about 40 MPs. With no public campaign event so far, he appears to be trying to keep a low profile and avoid media interrogation at least until he makes it to the final shortlist of two candidates.

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Jeremy Hunt and Michael Gove both have about 30 endorsements from Tory MPs and are vying for second place on the ballot, but a wide field of candidates is trailing them in the hope of gaining momentum in the next few weeks.

The hustings took place shortly after the Conservative party changed the rules of its contest to speed it up and knock out lesser-known candidates without enough support to reach the final round.

The party announced that hopefuls would need at least eight Tory MP supporters to stand, then 17 to get through the first round of voting by MPs and 33 to make it through the second round.

In subsequent rounds, the candidate with the lowest number of MPs supporting them will be knocked out until there is a final shortlist of two candidates.

The parliamentary stage is expected to take two weeks, with a final round of voting on 20 June. About 160,000 Conservative members will then cast their ballots, with an announcement of a new prime minister coming in the week beginning 22 July.