Andy Burnham said Labour has a "final chance to change" after his victory in the Makerfield by-election paved the way for him to challenge Sir Keir Starmer for the party leadership.
Burnham said his win could mark a "turning point" for the country, while he pledged not to use the constituency as a stepping stone.
But Prime Minister Sir Keir defended his government's record on delivering change and insisted he would stand in any leadership contest, adding: "I'm not going to walk away."
Burnham was joined by several Labour MPs at a victory rally on Friday, while his allies increased their calls for Sir Keir to stand aside and agree a "managed way forward" to avoid a bruising contest.
There were three by-election results overnight and in Aberdeen South, the Scottish Conservatives won a Westminster by-election for the first time in more than 50 years taking the seat from the SNP; while in Arbroath and Broughty Ferry the SNP held onto the seat in its by-election overnight.
Burnham won the Makerfield constituency after securing nearly 55% of the vote.
The former Greater Manchester mayor held off a challenge from Reform UK, which came second but more than 9,000 votes behind Labour.
Speaking to BBC Breakfast, Reform Deputy Leader Richard Tice said his party did not win as people had "voted Burnham to guarantee that the prime minister is ousted".
Sir Keir congratulated Burnham and claimed recent by-election results suggest the "tide is turning on Reform", telling broadcasters: "They've reached probably the peak of their support, it's going down."
He went on to argue his government was delivering on the "very considerable mandate for change" it secured in 2024.
In a lunchtime call with Labour staff members, Sir Keir said the party needs to "pull together" and "take the fight to Reform" in the forthcoming Greater Manchester Mayor election.
He went on to suggest a leadership contest could end up "tearing apart our party and our movement".
Addressing supporters at the election count, Burnham told voters he would not be turning away from the constituency as he headed to Westminster.
"Everyone knows that politics isn't working," he said. "Everyone can feel that the country isn't where it should be.
"Tonight could, just could, be a turning point."
He added: "It [Makerfield] will never be a stepping stone to me but instead will be my touchstone."
In a direct message to Labour MPs he said: "I do say to my own party: this is a final chance to change.
"This is what people said directly to me on the hundreds of doorsteps that I stood on.
"We must hear it, we must act upon it and we must get it right. There will be no second chance."
Burnham told his victory rally at Ashton Town FC later on Friday morning that his campaign was won by a "band of strong northern power women" – thanking Labour MPs Louise Haigh and Anneliese Midgley – but he steered clear of speaking about the prospect of a leadership challenge.
Burnham said voters had told him about the need to "make life more affordable", adding: "We do need to bring down water bills, energy bills, rail fares, just as we brought down bus fares in Greater Manchester."
He went on to speak of a desire for a "new drive of reindustrialisation" and raised people's concerns about the "unfairness of the immigration system".
Burnham added: "It is our last chance to change, but we're going to take it, aren't we? We are going to take that opportunity and we are going to lay out a new path for Britain."
Haigh said she hoped the PM and Burnham can "speak and agree a managed way forward".
Speaking to the BBC, the former transport secretary said "it's quite clear the prime minister can't take us into another set of elections" and that she hoped he "reflects on the results and does what's right in the interest of the country and the party".
Unite General Secretary Sharon Graham also said it was "clear that there now needs to be an orderly timetable for a leadership election".
And that call was echoed in the Labour party, with Polly Billington, MP for East Thanet, telling BBC Radio 4's World At One programme that she was "disappointed" to hear the prime minister talk of "fighting on" in a potential leadership election.
"That's not the language that we need and it's not the process that we need," she said.
"We need something that's both dignified for him, gives him the opportunity for a legacy, and gives us the opportunity to serve the people in the way we promised that we would two years ago."
Former Health Secretary Wes Streeting, who is expected to enter any leadership contest, wrote on X that Burnham's victory "gives us all hope that Labour can still win".
He added: "But Andy's campaign is proof that to do so we need to change."
The by-election came after Labour MP Josh Simons stepped down to allow Burnham an opportunity to contest the seat, after Labour's ruling body blocked his attempt to run in February's Gorton and Denton by-election.
Makerfield has been a Labour stronghold for 120 years but Reform were victorious in May's English council elections.
Nigel Farage's party was running high in the polls at the start of the campaign, but Robert Kenyon, a plumber and Reform councillor, ending up in second place with 15,696 votes.
Former Reform MP Rupert Lowe's rival Restore party ended in third place, with 3,111 votes for its candidate Rebecca Shepherd.
In contrast to Burnham's victory, Scottish Labour was left in fourth place in both by-elections north of the border.
Douglas Lumsden won Aberdeen South by a convincing margin for the Scottish Conservatives, beating the SNP, with Reform coming third and pushing Labour into fourth place.
In his victory speech, Lumsden told his supporters: "The destruction of the North Sea oil and gas industry must stop now."
The seat had been held by the SNP's Stephen Flynn since 2019 but has shifted over the years, with a Tory MP in 2017, SNP in 2015, and Labour's Anne Begg between 1997-2015.
Flynn's SNP colleague Stephen Gethins, who along with Flynn was elected to Holyrood in May, had to give up his Arbroath and Broughty Ferry seat at Westminster.
Here the SNP held onto this seat in the by-election, with Lara Bird winning by more than 5,000 votes.
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