Video obtained by APTN on Friday showed a river of mud running through a small alley in the village, called Atrani, tipping cars over and almost entirely covering some of them.
Officials said a small river close to the village had broken its banks, fed by heavy rains, on Thursday evening.
Brunella Cimadomo of the local Civil Protection Department said a 25-year-old woman was missing after the coffee bar where she worked was hit by the mudslide. She said the rain had let up on Friday but more was expected in coming days.
Atrani is less than a mile away from Amalfi itself.