Mr Corbyn visited the Cambridgeshire city to hail the victory of Labour’s Lisa Forbes as “a great win” which had “pushed the Tories to the margins” and shown widespread support for the party’s anti-austerity message.
Ms Forbes held the seat by a margin of 683 over Nigel Farage’s Brexit Party, which had been hotly tipped to land its first MP just weeks after being created.
Mr Farage slipped away from the count moments before the result was announced, but later said that his party’s close second place showed that UK politics had “fundamentally changed” and that Leave-backing voters should switch from the Tories to secure Brexit.
“It is no longer just two parties contesting,” he said. “The danger is that in seats like this the Conservatives split the Leave vote.”
In her victory address, Ms Forbes hailed the defeat of “the politics of division”
But there was unease in Labour ranks over the election of a candidate who had been forced to apologise for “liking” antisemitic messages on social media.
Former minister Dame Margaret Hodge, who has clashed with Mr Corbyn over antisemitism, said she had “seriously mixed feelings” about the result. She said she had formally raised concerns with the party leadership, adding: “Lisa Forbes and the Labour Party have a lot to answer for.”
The Jewish Labour Movement called for the new MP to have the whip suspended immediately.
The group said in a statement that Ms Forbes "will have to go far, far further than her statement at the weekend to demonstrate to our members and the Jewish community that she isn’t racist against Jews".
Ilford North MP Wes Streeting said he had “given up any hope that the Labour leadership is serious about tackling antisemitism, while Birmingham Yardley’s Jess Phillips said she would take Ms Forbes’ apology at face value”, but added: “With every case, the party’s values chip away and our ability to stand up against hate erodes.”
Mr Hunt said it was “incredibly disappointing” that Tories had slumped into third place.
He warned: “No future for our party until we deliver Brexit – any elections before then will just allow Corbyn to sneak through the middle. But when the UK has Brexited, we will be back.”
Mr Johnson said that if EU withdrawal was not delivered by October 31, “we risk Brexit Party votes delivering Corbyn to No 10”, while Mr Raab said any further delay would let Labour into power “by the back door”.
But international development secretary Rory Stewart – who has said he is ready to delay Brexit to deliver a deal – challenged his leadership rivals’ reading of the Peterborough result.
Figures from the British Electoral Survey showed that 88% of Conservative-held seats had seen a swing to Remain parties since the 2016 referendum, with 124 of them now containing more voters who support Remain-backing parties than no-dealers, he said.
“We must reach ch across divides to the centre ground and win back younger, Remain voters as well as Leave voters,” said Mr Stewart. “That’s the right and the smart thing to do. And I will do it.”
Labour MP John Mann, who wants Brexit delivered, said the Peterborough result showed that Mr Corbyn must resist internal pressure to commit the party to supporting a second referendum.
“The main conclusion to be drawn is that second referendum for the Labour Party is now dead,” he said. “We didn’t lose many votes to the Liberals. The Liberals got half the votes that they did in 2010. The thing that will stop the Labour Party winning the next general election is all this nonsense about a second referendum.
“We promised to deliver Brexit. That’s what we need to do, and if we do we will be in power.”
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