Holidaymakers trying to apply for their European health insurance card (Ehic) have been alarmed to discover that the online service is “currently unavailable”.

However, the NHS, which runs the service, told the Guardian that it was almost certainly a technical glitch rather than “anything to do with the referendum … It’s an unhappy coincidence”.

The Ehic gives access to state-provided healthcare during a temporary stay in another European Economic Area country, and Switzerland.

Those logging on to the official NHS Ehic website are currently being met with a message stating: “Service unavailable - The European health insurance card (Ehic) service is currently unavailable. Please try again later. Alternatively, you can apply by calling 0300 330 1350.”

However, when the Guardian called that number it was told that phone applications were also unavailable and that there was a system error.

The service has been unavailable for a number of hours, with no explanation given on the website. Some would-be applicants said they tried to apply on Sunday 26 June but were not able to access the service.

If the delays continue it could potentially pose a problem with last-minute applications from people about to set off on holiday.

Millions of people in the UK carry an Ehic, and there has been much debate following the referendum result about whether the cards will still be valid.

Some people expressed their frustration on Twitter. Emilie Renard wrote: “The bf and I just realised our EHIC cards have expired. Website is down, and the phone line just hangs up after being on hold. WTF #brexit”

Anna wrote: “...Been trying online and on the phone for two days to renew my EHIC.”

Meanwhile, Nick Lightbody wrote: “#EHIC currently unavailable - the effect of #brexit?”

THe Ehic scheme is free of charge and allows people to receive free or reduced-cost healthcare in other EU countries. It covers treatment that is “medically necessary until your planned return home”, though it should not be seen as an alternative to travel insurance.

An NHS spokesman apologised and said: “We’ve been experiencing problems with a number of our applications. We think it’s down to a hardware problem rather than anything to do with increased demand or anything to do with the referendum.” He added that engineers were looking into the issue, and the service was expected to be up and running again later on Monday afternoon.

The cards became one of the many battlegrounds during the run-up to the referendum. The official stated position of the remain campaign was that if the UK left, “our access to the Ehic would become uncertain. We would have to negotiate to be able to continue to use it”.

However, the leave campaign’s position was that “other non-EU states have the right to use the Ehic, and it will be in the interests of the rest of the EU as well as Britain to make sure we continue to offer healthcare to each others’ citizens”.

Switzerland and Iceland’s existing Ehic membership suggests it could well remain open to non-EU members, but it would have to be renegotiated, possibly with each individual country if some objected. The Association of British Travel Agents has warned that consumers would need to cover any additional health insurance costs through travel insurance should the UK exit the Ehic scheme.