Labour MPs have criticised Kemi Badenoch after a fractious prime minister’s questions on Wednesday, in which she accused the education secretary of being a “spiteful class warrior”.

In a rare move, Badenoch was chastised by the speaker, Lindsay Hoyle, in the chamber of the House of Commons after she took the personal swipe at Bridget Phillipson before saying that Keir Starmer had been betrayed and had “400 knives stuck in his back”.

“Let us think about the language we use,” said Hoyle. “Because when we leave this chamber, don’t be surprised when constituents feel they can use the same language. Let us show a little bit more decorum and respect.”

Phillipson shook her head as the Conservative leader – referring to the Labour policy of applying 20% VAT to private school fees from the start of 2025 – said she had “taxed private schools to pay for more teachers, but the number of teachers has gone down” and had let Starmer down with “her incompetence”. Badenoch added: “It turns out appointing a spiteful class warrior as education secretary was a disaster.”

Starmer responded that Phillipson, who experienced childhood poverty growing up in the north-east of England, was an “incredible story of social mobility and success” and added: “I am so proud … It drives every single priority and value that she has.

“I would have thought the party opposite would recognise and understand some of that, but they’ve fallen so low, they don’t.”

Badenoch then turned on cheering Labour MPs, accusing them of “cheering so loudly with their 400 knives stuck in [Starmer’s] back”. When MPs reacted angrily, she said: “They don’t like it up them, but they know what I’m saying.”

After PMQs, sources reported a row broke out between Badenoch and Phillipson as MPs left the chamber. Labour sources said the technology secretary, Liz Kendall, said to Badenoch as they were leaving: “What you said was absolutely outrageous,” at which Badenoch turned to Phillipson and said: “You are spiteful. I will keep talking about how spiteful you are.”

An ally of Phillipson said Labour whips were referring the incident to the speaker, while the Labour MP Tulip Siddiq said she would make a point of order, a procedure used to draw attention to a potential violation of parliamentary rules.

“Badenoch has yet again disgraced herself,” said an ally of Phillipson. “She’s chosen to attack the only working-class woman from the north-east of England in the cabinet, because the Tories hate working-class people who do well. If standing up for the 94% of kids in state schools makes Bridget a ‘spiteful class warrior’, then she’ll wear it with pride.”

Conservative sources disputed this version of events and said Phillipson had “aimed a barb” at Badenoch as they were leaving the chamber, to which the leader of the opposition said: “I’ll fight you all the way. You’re destroying children’s lives.”

Asked by the Guardian if Badenoch would apologise for comments made in the chamber following the rebuke, the Tory leader’s spokesperson said “absolutely not”. A Conservative source said: “The fact is that the vindictive and class war tax hike on independent schools has forced the closure of several of them, disrupting children’s lives, forcing them into state schools in the middle of their studies, putting further pressure on the state school sector, and we are absolutely determined to oppose it all the way.”

Badenoch later posted on social media: “‘I grew up on a council estate’ is not an excuse for failure. You are sacrificing the future of generations of kids on the altar of your class envy – reversing even Labour’s academy reforms.”

Following the altercation, senior Labour figures defended the education secretary. David Lammy said the attack laid bare “something deeper about the direction of their politics”. The deputy prime minister said: “I’m proud that this Labour government has the most state-schooled cabinet in the postwar era – people who didn’t grow up with privilege, but earned their place through hard work.”

The Scotland secretary, Douglas Alexander, said Badenoch’s words “said a lot more about her than Bridget”. He added: “Like Bridget, I’m proudly state-schooled: we’re the most state-schooled cabinet in the postwar era. We’re not motivated by spite but by tackling poverty and extending opportunity.”