Update 21.3.2020 Published in the State Bulletin today: Driving permits and licenses that expire during the State of Alarm are automatically extended for up to 60 days after it ends. The same goes for temporary ones. Holders of licences that expired during the State of Alarm will not be fined.
The 6 months that holders of non-EU driving licences can drive here maximally, will be suspended for the duration of the State of Alarm.
Automated translation from Spanish press release, published on the Ministry of Interior webpage.
Press Release with new preventive measures adopted by the DGT as a consequence of COVID19