The result - in a seat Labour has held for nearly 100 years - has renewed criticism of Sir Keir's leadership from Labour MPs.
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Sir Keir Starmer has vowed to "keep on fighting" after Labour suffered a devastating by-election defeat to the Green Party in Gorton and Denton.
Labour, which took the Greater Manchester seat with more than 50% of the vote in 2024, was pushed into third place behind Reform UK.
The prime minister said the result was "disappointing" but he would "fight against extremes in politics" on both the left and the right that "want to tear our country apart".
Green leader Zack Polanski said the victory showed there were no longer any "no-go areas" for his party, as winning candidate Hannah Spencer said "this is only the beginning".
The result - in a seat Labour has held for nearly 100 years - has renewed criticism of Sir Keir's leadership from Labour MPs, with some calling for him to stand down.
It piles further pressure on the PM ahead of crucial May elections in Scotland, Wales and some English councils.
Former Deputy Prime Minister Angela Rayner - a possible challenger to Sir Keir - said it "must be a wake up call" and the government must be "braver".
Others blamed a decision to block Greater Manchester mayor Andy Burnham, another potential leadership candidate, from standing in the seat by Labour's ruling National Executive Committee.
The move was backed by the prime minister but some Labour MPs argued Burnham, who is hugely popular in the region, would have been the party's best chance of winning the by-election.
Sir Keir said: "Incumbent governments quite often get results like that mid-term, but I do understand that voters are frustrated. They're impatient for change."
Asked if he had considered resigning, he said: "I came into politics late in life to fight for change for those people who need it...
"I will keep on fighting for those people for as long as I've got breath in my body."
Challenged over whether he was wrong to block Burnham from standing, the PM said: "We had an excellent candidate, absolutely rooted in her community."
The Greens' victory marks a breakthrough moment for the party, delivering its first-ever MP in the north of England.
Since taking charge of the Greens five months ago, self-described "eco-populist" Polanski has courted voters with a left-wing pitch on the economy, alongside its traditional focus on environmental issues.
Its policies include giving councils the power to control rents, extending free school meals to all children, and a imposing a new 'wealth tax' on assets above £10m.
But in a letter to Labour MPs, Sir Keir said the party had capitalised on an endorsement from George Galloway's Workers Party of Britain – which won 10% of the vote in the constituency in 2024 but did not stand in this by-election - and his "divisive, sectarian politics".
The PM also attacked the Green Party policy of legalising all drugs and said they "simply do not have the resources, the activist base or the local knowledge to replicate this victory across the country".
Spencer dismissed accusations of sectarian politics, insisting her party had united voters with commons concerns about the cost-of-living, public services and the war in Gaza, whilst Polanski accused Starmer of trying to "smear voters as extremists".
Conservative leader Kemi Badenoch said the result "shows Keir Starmer's premiership is finished" and the PM would resign "if he had any integrity".
The by-election was triggered by the resignation of former Labour health minister Andrew Gwynne, who was suspended from the parliamentary party for offensive WhatsApp messages a year ago.
While polls had suggested a close race, the Green Party ultimately won comfortably, with nearly 41% of votes cast and a majority of more than 4,000.
The Conservatives came fourth with just 706 votes - the party's worst ever by-election result.
So far public condemnation of Sir Keir following the result has largely been confined to frequent critics of his leadership.
Labour MP Karl Turner said the result was "catastrophic", as he blamed the decision to block Burnham as a candidate and "unpopular" policies.
Another MP, Richard Burgon, who is on the left of the party, said blame for the defeat "lies squarely with Keir Starmer and his clique", who "put factional interests over having the candidate best placed to win".
"If Labour is to be the 'Stop Reform' party, then the leadership must stop treating progressive voters with contempt - and start appealing to them," he said.
Fellow left-wing MP Nadia Whittome said her party needed "change at the top" and "should be true to the progressive values that Labour is meant to stand for".
However, more MPs told the BBC on the condition of anonymity the PM should stand down.
One described the by-election result as "a punch in the face for the Labour Party and for Keir Starmer's premiership".
"Keir must reflect on how unpopular he is with the public," another said.
"Cabinet ministers might be doing good interviews on the airwaves in tough circumstances but they're like cabin crew successfully calming passengers in a plane that's crashing.
"If they manage to calm things down, the plane is still crashing. And who is in the cockpit is what matters."
Others advised caution, with one Labour MP telling the BBC: "We need to avoid knee-jerk responses that address the concerns of one wing of the support we need at the expense of losing the other, and focus on the core agenda around tackling the cost of living crisis and improving public services which all the voters we need to win back would share."
Another said there was "no obvious alternative" to Sir Keir and "a leadership contest will be disastrous".
Spencer - a plumber and councillor in Trafford - becomes the Green Party's fifth MP.
She said the result - in what the party has said was its 127th target seat - showed the Greens "can win anywhere".
Spencer added that her party's first ever victory in a Westminster by-election was "just the beginning", as she predicted further gains in elections to English councils and the Welsh Senedd in May.
Polanski said it was "an existential crisis" for Labour, which had sought to present the by-election as a two-way fight with Reform UK but ended up in third place.
"No longer can they try and scare people into saying they have to vote for something because they're worried about the least-worst option," the Green Party leader added.
Reform UK candidate Matt Goodwin said his party had "embarrassed Labour in one of their strongest seats".
He described the Green victory as a "coalition of Islamists and woke progressives", marking "the emergence of a dangerous sectarianism in British politics".
