On many different levels the British government and the European Union are working on the details around the Brexit. This will have a lot of impact on British companies and persons, maybe also on the topic of domain registrations.
Registry EURid manages the .eu and .ею extensions under mandate of the European Commission. This mandate defines that the registrant of a .eu domain must reside within the European Union, which may be a problem after the Brexit is effective.
The latest information that we have, is that the Commission has asked EURid to prepare domain name revocation scenarios. Note that nothing has been decided yet: EURid is in close contact with the Commission to reach a solution that is optimal for all parties.
Until further notice, British registrants will be able to register and renew their .eu domain names. We will inform you through our newsletters if new information is available. Find a formal statement from EURid on this matter on EURid’s website.
The same question applies to other registries that require a presence in the European Union, like AFNIC (.fr) and NIC IT (.it). Those registries have informed us that they do not decide anything yet before more details on the Brexit are available.