Around 10,000 EU nationals have quit the NHS since the Brexit referendum, it has emerged.

NHS Digital, the agency that collects data on the health service, found that in the 12 months to June, 9,832 EU doctors, nurses and support staff had left, with more believed to have followed in the past three months.

This is an increase of 22% on the previous year and up 42% on two years previously. Among those from the EU who left the NHS between June 2016 and June 2017 were 3,885 nurses and 1,794 doctors.

This is the first time anecdotal evidence of Brexit fallout for the NHS has been quantified. The staff losses will intensify the recruitment problems of the NHS, which is struggling to retain nurses and doctors.

The British Medical Association said the findings mirrored its own research, which found that four in 10 EU doctors were considering leaving, with a further 25% unsure about what to do since the referendum.

“More than a year has passed since the referendum yet the government has failed to produce any detail on what the future holds for EU citizens and their families living in the UK,” said a spokeswoman for the BMA, which represents 170,000 doctors. “Theresa May needs to end the uncertainty and grant EEA [European Economic Area] doctors working in the NHS permanent residence, rather than using them as political pawns in negotiations.”

The Liberal Democrat leader, Vince Cable, called on May to take urgent action to stop a further exodus at the NHS. “Theresa May must make a bold offer to the EU to ringfence negotiations on citizens’ rights and come to a rapid agreement. We are losing thousands of high-quality nurses and doctors from the NHS, driven partly by this government’s heartless approach to the Brexit talks,” he said.

He and others, including Tory MP and former attorney general Dominic Grieve, want the issue of EU citizens ringfenced from the main Brexit talks to help staunch a potential exodus of valuable EU workers from Britain.

“It’s time for the government to take the issue of EU citizens in the UK and British nationals in Europe out of these negotiations,” said Cable.

This year it emerged that 40,000 nursing posts were now vacant in the NHS in England as the service heads for the worst recruitment crisis in its history, according to official new data.

The figures came a week after a German consultant at a Bristol hospital told the Guardian how his department would be “closed down” if its Spanish nurses did not stay. Peter Klepsch, an anaesthetist, told how he and his neuroscientist wife came to the UK 12 years ago and had planned on staying but were now questioning their long-term future here.

“I know a lot of EU doctors and nurses who are saying, ‘I’m not going to stay very much longer,’” Klepsch said during a day of protest organised by the3million grassroots campaign for EU citizens.

The government has been accused of failing to deliver on its promise to guarantee EU citizens’ rights post Brexit and instead using them as a bargaining chip in negotiations.

In May the UK offered to guarantee the rights of all EU citizens affected by the UK’s decision to quit the bloc, including the 1.2 million British nationals already settled or retired on the continent. However the offer was condemned as “pathetic” by the3million. This has led to an impasse in Brexit negotiations, with less than half the listed issues agreed after the third round of talks ended in August.

• This article was amended on 22 September 2017. An earlier version said the EU offered to guarantee the rights of all EU citizens in May. This has been corrected to say the UK made an offer.

