England and Wales’s most senior judge has written to Keir Starmer about an “unacceptable” exchange with Kemi Badenoch at prime minister’s questions, saying she was “deeply troubled” by the discussion on the case of a Palestinian family’s right to live in the UK.

Lady Sue Carr, the lady chief justice, criticised the Conservative leader’s questions about the case, in which a family from Gaza had applied through a scheme designed for Ukrainian refugees.

She said it was also unacceptable for the prime minister to respond by saying the decision had been wrong and that the home secretary would be “working on closing this loophole”.

Starmer’s spokesperson said it was clear from his past as director of public prosecutions that he had the utmost respect for the independence of the judiciary.

“The prime minister has made clear that it is for parliament to make the laws and for the government to decide policy. Where the law is not working as we think it should be, the government will take action to tighten up the rules – and that is what we are doing,” they said.

Badenoch hit back at the criticism from the lady chief justice, saying it was essential politicians could debate the outcome of such cases. “Parliament is sovereign. Politicians must be able to discuss matters of crucial public importance in parliament,” she said.

The exchange last week referenced reports of an appeal by the family against the decision by an immigration tribunal judge in September to dismiss their claim. A further appeal was allowed by upper tribunal judges in January.

Carr said she was “deeply troubled to learn of the exchanges” at PMQs and told reporters she had written letters about her concern regarding judicial decisions. “I think it started from a question from the opposition suggesting that the decision in a certain case was wrong, and obviously the prime minister’s response to that. Both question and the answer were unacceptable,” she said.

“It is for the government visibly to respect and protect the independence of the judiciary. Where parties, including the government, disagree with their findings, they should do so through the appellate process.”

Carr said she had written to Starmer and to the justice secretary, Shabana Mahmood.

The case involved a family with four children whose home in Gaza was destroyed by an Israeli airstrike. They applied for entry to the UK using the Ukraine Family Scheme to join the father’s brother, who has lived in the UK since 2007 and is a British citizen. That application was first refused in May last year after the Home Office concluded the requirements of the scheme had not been met.

An upper tribunal judge then allowed the family to come to the UK on the basis of their right to a family life under the European convention on human rights (ECHR).

Starmer told the Commons: “I do not agree with the decision. She’s right, it’s the wrong decision.

“But, let me be clear, it should be parliament that makes the rules on immigration. It should be the government that makes the policy, that is the principle, and the home secretary is already looking at the legal loophole which we need to close in this particular case.”

Badenoch wrote in a post on X that there was nothing wrong with her line of questioning. “This doesn’t compromise the independence of the judiciary. The decision to allow a family from Gaza to come to the UK was outrageous for many reasons,” she wrote.

The shadow home secretary, Chris Philp, said politicians were “perfectly entitled to comment on decisions by judges”. The shadow justice secretary, Robert Jenrick, also said it was wrong. He said: “The principle of the rule of law is being misused. It needs to be reclaimed. It does not, and never has meant, rule by lawyers.”

Carr’s intervention also prompted criticism by some senior legal figures. Richard Ekins KC, who heads the Judicial Power project for the rightwing thinktank Policy Exchange, said it was “a very ill-advised intervention”.

He said: “There was nothing in the least constitutionally improper in the recent exchange between the prime minister and the leader of the opposition. Neither judicial independence nor the rule of law entitle judges to be free from criticism and the lady chief justice is wrong to attempt to suppress criticism.”

The former justice secretary Robert Buckland told the Guardian the context of the remarks was the climate of personal attacks in which many judges were operating. “It is certainly not the case that politicians cannot disagree with or criticise judicial decisions, but it has to be done responsibly,” Buckland said.

Speaking at her annual press conference, Carr made a wider point about the public safety of the judiciary. Senior judges were often directly criticised by the Conservative administration – particularly the supreme court’s interventions on Brexit – which Starmer and Labour criticised them for in opposition.

Carr said: “It is not acceptable for judges to be the subject of personal attacks for doing no more than their jobs. Their job is to find the facts on the evidence before them and apply the law as it stands to those facts.”

She said matters had come to a very “dramatic and concerning head” with the attack on Judge Patrick Perusko, who had a radiator thrown at him in 2023.

It is understood the government intends to respond to Carr’s letter shortly but has not yet done so. The home secretary, Yvette Cooper, has been tasked with looking at the application of article 8 of the ECHR in this case.