Defiant peers have again dug in their heels over Rishi Sunak’s Rwanda deportation plan and demanded MPs think again for a fifth time.

The House of Lords backed by 240 votes to 211, majority 29, a requirement that the east African country could not be treated as safe until the secretary of state, having consulted an independent monitoring body, made a statement to Parliament to that effect.

It means a continuation of the parliamentary tussle over the Safety of Rwanda (Asylum and Immigration) Bill, known as “ping-pong”, where legislation is batted between the two Houses until agreement is reached.

However, the opposition did not press its demand for the Bill to include an exemption from removal for Afghan nationals who assisted British troops after what critics hailed as a concession.

A Home Office minister said the Government will not send those who are eligible under the Afghan Relocations and Assistance Policy (Arap) to Rwanda.

Lord Sharpe of Epsom told peers: “Once this review of Arap decisions for those with credible links to Afghan specialist units has concluded, the Government will not remove to Rwanda those who received a positive eligibility decision as a result of this review where they are already in the UK as of today.”

Labour former defence secretary Lord Browne of Ladyton, who had been leading calls for such an assurance, said: “The minister does not believe this to be a concession, it is to him a restatement of what he has been telling us for some time, but in a different form.”

However, pointing out it would now mean people not being removed, Lord Browne argued “that is a concession in anybody’s language”.

He added: “And it’s an extremely important concession, because these are the small number of people that… I have said are the target of my ambition that they will not be deported.”

The Bill and a treaty with Rwanda are intended to prevent further legal challenges to the stalled asylum scheme after the Supreme Court ruled the plan was unlawful.

As well as compelling judges to regard the east African country as safe, it would give ministers the power to ignore emergency injunctions.

The Prime Minister says this will act as a deterrent to migrants attempting to cross the English Channel in small boats.

Earlier on Monday, Mr Sunak blamed Labour peers for holding up the Bill, as he acknowledged he will miss his self-imposed spring target for getting the Rwanda scheme off the ground.

Speaking in the Lords, leading lawyer and independent crossbencher Lord Anderson of Ipswich said: “We are in the end game now. We will, this week, have a law that provides for the offshore processing and settlement of asylum seekers in Rwanda.

“Its benefits remain to be seen. It costs will be measured, not only in money, but in principles debased: disregard for our international commitments, avoiding statutory protections for the vulnerable, and the removal of judicial scrutiny over the core issue of the safety of Rwanda.”

Lord Sharpe said: “I do not believe we have debased our principles, I believe we have upheld them.

“We have upheld the principle of the integrity of our sovereign borders, the principle of not ceding our immigration policy to criminal gangs, the principle to safeguard lives and deter dangerous and illegal Channel crossings. That is the point of this Bill.”