A wave of muddy water engulfed the town of Sant Llorenç des Cardassar, in the east of the island, after heavy rain caused a river to burst its banks.
Vehicles were swept away as water gushed through narrow streets.
Hundreds of emergency workers and sniffer dogs are involved in the search-and-rescue effort. A five-year-old child is missing.
The dead include six Spanish nationals, a Dutch tourist, a British couple and one unidentified victim.
Worst affected was the town of Sant Llorenç des Cardassar, where cars were swept away. The British couple was travelling in a taxi there when it was caught in the flood. The driver was among the victims.
A woman who had been in her car with her two young children also died. As the waters rose, she managed to get her eight-year-old daughter to safety, but she was unable to save her son aged five, who is still missing.