In Slovakia, you must carry your passport or EU national identity card with you at all times. You will need it for almost everything – from applying for health insurance to getting a bus pass or a pay-as-you-go SIM card. You also need it to enter certain museums and office buildings. As far as I remember, the only time I didn’t need to show my passport was when I applied for my Tesco clubcard.
Driving licences and credit/debit cards are not acceptable forms of ID in Slovakia. You can use your Trvalý pobyt (residence permit) as an ID in this country.