Theresa May has been urged to maintain Britain’s membership of EU law enforcement agencies such as Europol after Brexit.

The calls came from Sir Hugh Orde, the former chief constable of police in Northern Ireland, Max-Peter Ratzel, former head of Europol, and Dominic Grieve, the Conservative chairman of the parliamentary Intelligence and Security Committee.

They said the fight against terrorism made it crucial to maintain close co-operation in the EU’s security and intelligence agencies, even if it required some continued involvement with the European Court of Justice.

“Although our partnership with the US for intelligence sharing is extremely important, the fact is that the current terrorist threat is very much a European dimension issue,” Mr Grieve told The Observer.

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“The Schengen database and knowing about who has moved where are all intimately dependent on European systems and we have got to try to remain in them.

“I think it is going to be very difficult, and the Government and Parliament will have to face up to the fact that it might turn out during the Brexit negotiation that some sort of mechanism for the European court is going to be necessary.”

Mr Orde said UK membership of EU bodies such as Europol and Eurojust, which negotiates judicial cooperation in criminal cases, allows access to huge amounts of vital data and allows police in the UK to set up joint enquiries with those from other national forces without delay.

He said European arrest warrants were also essential, adding: “If we don’t have all this, it makes it a lot more difficult to do this crucial work. It is vital that we get to a situation as close to what we have as members of the EU as possible, though it is difficult to see how we do that.”

Mr Ratzel warned Britain must accept some ECJ rulings as part of the deal, which Ms May has previously ruled out.

He said: “My expectation is that once you have an important role to play, you have to be under the European Court of Justice. Cherry picking cannot work. You have to obey the rules of the club. I am personally sure that the Europeans will never accept Britain having a role in Europol without being under the rule of the European Court of Justice.”

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The Home Secretary, Amber Rudd, previously said it is “likely” the UK will leave Europol in the wake of Brexit unless a deal is made.

She said Britain could take its data away from the police and security cooperation organisation unless there was a deal.

It comes as Ms May proposed a new Commission for Countering Extremism, which will have a remit to clamp down on “unacceptable cultural norms” such as female genital mutilation.

It will also act to ensure that women’s rights are upheld in all of Britain’s ethnic and religious communities, the Prime Minister said.

Ms May made clear that the commission will be expected to stand up for women’s rights as extremism often goes hand in hand with poor treatment of women and girls.

She said: “Britain is one of the world’s most successful multi-racial, multi-religious and multi-cultural societies.

“But our enjoyment of Britain’s diversity must not prevent us from confronting the menace of extremism, even if that is sometimes embarrassing or difficult to do.

“Extremism, especially Islamist extremism, strips some people of the freedoms they should enjoy, undermines the cohesion of our society, and can fuel violence. And it can be especially bad for women.

“There is clearly a role for Government in tackling extremism where it involves behaviour that is or ought to be criminal.

“But there is also a role for Government to help people and build up organisations in society to promote and defend Britain’s pluralistic values, and stand up to the extremists who want to undermine our values and impose their twisted beliefs onto the rest of us. That is what this plan is all about.