official site for visitors to Bratislava  

To be valid for entry, a passport must be valid for the duration of one's stay in Slovakia, and for 2 months after the end of any visa required for entry. Visas can be obtained from whichever Slovak embassy is accredited to the visitor's country.

The Slovak Republic is a member of the EU and a full member of the Schengen Area. EU citizens have the right to move freely within and between other EU countries and have the right to stay in another EU country for a 6-month period providing they hold a valid passport or national identity card.

If you lose your passport, this should be reported to the Foreigners’ Police (Cudzinecká polícia)
Hrobákova 44, in the Petržalka district; EU citizens tel: +421 (0)961 036-871;
non-EU citizens tel: +421 (0)961 036-866 or (0)961 036-867.

Your embassy or consulate can normally issue a temporary passport.