At least 10 people, including two Britons, have been killed after torrential rain caused flash flooding on the Spanish island of Mallorca.

The British citizens, thought to have been a couple, were reported to have drowned in a taxi as flood water engulfed the town of Sant Llorenç des Cardassar, about 40 miles east of the capital, Palma, on Tuesday evening. Their deaths were confirmed by the Foreign Office on Wednesday afternoon.

“We are supporting the family of two British people following their deaths in Spain, and will do all we can to assist them at this deeply difficult time,” said an FCO spokeswoman. “Our staff remain in contact with the Spanish authorities who are responsible for responding to the floods, and are ready to assist any other British people who require our help.”

A spokeswoman for the Balearic islands’ emergency services said a Dutch woman had also died in the flooding. Other victims were found in Artà and the coastal town of S’illot.

Local emergency services, whose rescue efforts were supported by specialist military and police units, said 10 people had been killed in the floods. By late afternoon on Wednesday, one child was still missing.

The Spanish prime minister, Pedro Sánchez, flew to the island to offer support to the emergency services, and announced that Sant Llorenç would be declared a disaster zone. He offered his condolences to the victims’ families and said the area “can count on the government’s support in getting back to normal as soon as possible”.

Spanish weather forecasters said 20cm (8in) of rain fell in four hours.

“It all happened in less than 10 minutes,” an unidentified witness with wet clothes and a blanket over his shoulders told the public broadcaster TVE. “We had to swim to try to survive.”

Videos showed cars being washed away in fast-moving, muddy water more than a metre deep. Footage of the aftermath in Sant Llorenç showed dozens of damaged and upturned vehicles, many of them covered in debris.

Officials said more than 600 rescuers and medical staff were at work and that emergency relief centres had been set up in sports halls.

The tennis player Rafael Nadal, who is from Mallorca, offered to open up his sports centre and tennis academy to people displaced by the floods. “Our most sincere condolences to the loved ones of the victims of the serious floods in Sant Llorenç,” he wrote on Instagram.