Since Sunday, British nationals who have lived in the country for more than six months are no longer able to use DVLA-issued licences, as had been accepted before Brexit.
British residents will need to obtain a Spanish licence to drive, and may even need to retake a driving test until an agreement is reached.
Tourists are not affected by the move, who can still use their British licences.
A UK government source told the Mail on Sunday that the issue would be “temporary” and that talks were being accelerated to reach an agreement on the issue.
One British expat caught up in the problem said on social media: “That’s us screwed … I have no idea what we are going to do, we live up in the mountains in the middle of nowhere. Why on earth is this so hard?”
Another added on Twitter: “UK driving licenses now not valid for driving in Spain!
“Means I can’t go to work, nor can I continue my work with a dog charity here in Tenerife!”
A spokesperson for the Foreign Office told the paper: “An agreement to swap UK driving licences for Spanish licences has not yet been reached.
“From May 1, UK licence holders who have been resident in Spain for more than six months and did not exchange their licence during the transition period will no longer be able to drive legally in Spain.
“We have agreed to rapidly accelerate talks and are urging the Spanish Government to bring forward interim measures.
“We remain committed to reaching a deal and the UK already has similar arrangements in place with 24 other EU countries.”