EU citizens (including citizens of EEA countries) can use their driving license issued in other member state (if valid). If driving licence expire its validity you can renew it in country of your usual residence (including Slovakia).
Foreigners as tourists up to 90 days may use their driving licenses (issued in their country of origin). Non EU/EEA citizens that obtain temporary or permanent residency permits in Slovakia must apply for driving license in Slovakia (within 60 days after 185 days since residency permit had been issues - required by legislation). Driving licences administration is responsibility of state police. You can process your request at state administration Client Centre on Tomášikova Street in Bratislava (serving all citizens living in Bratislava).
State administration Client Centre opening hours:
Address: Tomášikova street 46, Bratislava
Monday, Tuesday, Thursday 8:00 to 15:00
Wednesday 8:00 to 17:00
Friday 8:00 to 14:00