Of the lies told during the Brexit referendum – and there were many – perhaps the most egregious was the claim that we could spend an extra £350m on the NHS as a result of leaving the EU. It has gained unique notoriety not simply because the figure was demonstrably false, or even because Brexit will shrink the economy rather than free up vital funds, but rather because of its calculated emotional manipulation. We value the NHS more than any other institution. As the defining icon of the post-war consensus and intrinsic component of our national story, it unites Britons across political, geographical and class divides. Crippled by austerity, staff shortages and low morale, our NHS is also on its knees. But far from offering a helping hand, Brexit threatens to bring it down altogether.

A report in the Lancet offers a comprehensive – and bleak – analysis of the dangers. Brexit stands to damage staffing, funding, access to new products and technology, and standards of public health. The softer the Brexit, the lower the harm – but as Theresa May’s speech in Florence made clear, the government still plans to leave the single market, customs union and other EU bodies after a transition ends in 2021, no matter the cost.

Telling NHS workers they can help us, but forget about ever settling or becoming British, may not prove attractive

The key area of risk is also the central plank of Brexit: restrictions on free movement of people. This is no coincidence. While millions of leave voters expressed the concern that immigration was posing an intolerable burden on public services, studies have repeatedly indicated that it in fact keeps them afloat. The NHS and adult social care employs 150,000 EU nationals; 10% of our doctors graduated in EEA countries. The government continually promises that the “brightest and best” will always be welcome, but this elitist and divisive slogan fails even on its own terms. Britain’s most vulnerable patients do not simply depend on EU surgeons, GPs and nurses, but on an army of notionally “unskilled” carers, porters and cleaners who help to keep people alive.

Even if the government prioritises NHS workers in its post-Brexit immigration strategy, grave damage has already been done. This week, a molecular biologist in Madrid told me that London was his favourite city, but its political climate now too hostile to consider returning. The figures bear out the anecdote: while 40,000 nursing positions currently lie vacant, the number of EU nurses registering to work here has dropped by a staggering 96%. While fewer arrive, more depart. About 10,000 EU nationals have left the NHS in the past year.

Britain no longer feels like a welcoming place for foreigners. Let alone the shame, we should also feel profound alarm. We do not have the doctors and nurses that we need as it is; and even if the government was adequately investing in training – which it isn’t – we would still have no time to replace those Europeans who either intend to leave or never even come. To add idiocy to injury, the recently leaked government proposals on immigration specified time-limited work permits, with permanent residency a possibility only for the most highly skilled. Telling NHS workers that they can help us for a few years, but probably forget about ever settling or becoming British, may not prove an attractive offer.

The problem for the NHS is that unlike, say, the single market or Irish border issue, it is not in itself an EU competence and will not be negotiated at the Brexit table. What we do with our healthcare has always been a matter for us alone. But as with so much else in Brexit, problems both predictable and previously unforeseen are threatening key aspects of our national infrastructure.

While remain campaigners stressed the risks to the NHS of reduced immigration and a diminished economy, few mentioned the €3.5bn supplied by the European Investment Bank to the NHS since 2001, or publicised the dangers to cancer patients of leaving the European Atomic Energy Community or the European Medicines Agency. The government, for its part, is so consumed with fire-fighting that it is neglecting to recognise the NHS for what it is: one of Brexit’s key issues, and potentially its most high-profile piece of collateral damage.

Like the ravens at the Tower of London whose departure, in legend, presages the nation’s fall, the NHS’s success – or collapse – is also Britain’s. Brexit’s architects knew that people would respond to appeals to help it; faced with a false prospectus, the public duly chose British hospitals over Brussels bureaucrats. Those same voters may yet punish Brexit’s leaders, but the national consequences will profoundly eclipse any political ones. After all, the risk of deploying your most treasured family heirloom as a political football is not just that it could ultimately land in your own goal – but that in your recklessness, you may irreparably smash it.

• Jonathan Lis is deputy director of the thinktank British Influence