Education Secretary Bridget Phillipson has told the BBC she will be turning Kemi Badenoch’s attack on her at Prime Minister’s Questions into a T-shirt.

The Conservative leader called Phillipson a “spiteful class warrior” for taxing private school fees, claiming this was to pay for more teachers in state schools “but the number of teachers has gone down”.

The two women exchanged harsh words after PMQs and continued their row on social media, before both doubled down again on Thursday.

Asked whether she would apologise to Phillipson, Badenoch said she had also called the education secretary incompetent in the Commons and it was “interesting” that she had not complained about that.

Speaking on a visit to an army barracks in Essex on Thursday morning, the Tory leader told reporters Phillipson “could dish it out but can’t take it”.

At PMQs the previous day, Badenoch had attacked Phillipson for Labour axing the historic VAT exemption for private schools and introducing a 20% rate, which was expected to raise £1.8bn for the education budget, but was not ringfenced for hiring teachers specifically.

However, she was correct to point out that Labour remains 2,000 teachers short, external of its pledge to recruit 6,500 extra teachers, according to the latest government statistics.

The Tory leader also highlighted a poll this week from 4,000 National Education Union members, which showed 0% of respondents, external thought Phillipson was doing “very well” in the role, with 74% responding either quite or very badly.

Asked about Badenoch’s comments by Nick Robinson on BBC Radio 4’s Today programme, Phillipson said: “Next time you see me, Nick, I’ll be wearing a T-shirt saying ‘spiteful class warrior’ – because if being a spiteful class warrior means lifting half a million children out of poverty I’ll be wearing that T-shirt with pride.”

The education secretary was referring to Labour’s plan to tackle child poverty, external, which included scrapping the two-child benefits cap, expanding free childcare and free school meals, and creating 3,000 extra nursery places.

Phillipson said Badenoch had recently compared her to a Gestapo officer and added: “Kemi Badenoch can speak for herself and her own unique brand of unpleasant politics – I’m focused on better life chances for children.

“I think you’re losing the argument when you reduce yourself to that level of abuse.”

Deputy prime minister David Lammy has defended Phillipson in the row and Scottish Secretary Douglas Alexander said: “Like Bridget, I’m proudly state-schooled: we’re the most state-schooled cabinet in the post-war era.

“We’re not motivated by spite but by tackling poverty and extending opportunity.”

Phillipson was pressed on whether she had crossed a line herself, by accusing Conservative shadow justice secretary Nick Timothy of racism for describing a mass Muslim public prayer in Trafalgar Square as an “act of domination and division”.

In March, Sir Keir had called on Badenoch to sack Timothy for the comments at PMQs, claiming they showed her party had a “problem with Muslims”.

Phillipson had reignited that row ahead of PMQs on Wednesday morning’s education questions, external, in response to a question from shadow women and equalities minister Claire Coutinho raising concerns about the policing of criticism of Islam.

The education secretary responded: “We have seen shocking examples of Muslims in our country being targeted on the basis of their faith and because of who they are.

“We need look no further than [Nick Timothy], who engaged in appalling racism towards Muslims in our country and, rather shamefully, was not sacked by the Leader of the Opposition for those comments.”

Asked by Robinson whether labelling Timothy racist was hypocritical when she wanted to tone down political discourse, Phillipson said: “It was racist, he should be ashamed of himself, and he should have been sacked.

“If you want robust political debate, I’m here for that every day of the week, but I think reducing this to Nazi analogies, reducing it to that level of highly personalised abuse, I think says more about Kemi Badenoch than anybody else”.

Timothy denies his comments are racist and, on Thursday Badenoch went further still, demanding Phillipson apologise for calling Timothy a racist just before PMQs.

Asked whether she would apologise to Phillipson, she steadfastly refused to do so, instead pointing out that neither Phillipson nor Sir Keir had complained that the Tory leader had called the education secretary incompetent.

She said: “The reason why those taxes were put in place was to fund more teachers.

“Phillipson promised 6,500 teachers — we have 2,000 fewer teachers. We have gone backwards both years she has been education secretary.

“That’s a failure and it’s my job as the leader of the opposition to hold her to account.”

Pressed on whether politicians had a responsibility to set the tone of debate, Badenoch insisted there was a need for robust debate and called on Labour to stop complaining about “hurty words”.

Sign up for our Politics Essential newsletter to read top political analysis, gain insight from across the UK and stay up to speed with the big moments. It’ll be delivered straight to your inbox every weekday.