Three family members who swam to a private island when their boat caught fire and sank were rescued two days later after their marine radio washed ashore.

“Mayday, mayday, mayday! Our ship went down at Tarpaulin’s Cove!” Tyler Sullivan told a Coast Guard dispatcher early Wednesday morning. “Our ship burned while we were sleeping, and we barely escaped!”

Sullivan and his parents left Falmouth on Friday and had planned to return Tuesday after spending the weekend anchored close to Naushon Island, the largest of a chain of islands between southeastern Massachusetts and Martha’s Vineyard.

A relative contacted authorities Tuesday night after they failed to return as planned.

That triggered a multiagency search that included Falmouth police and the Falmouth Harbormaster.

In mayday recordings released by the Coast Guard, he described their location and what happened to the boat.

“The vessel has sunk, and we are in the farmhouse,” Sullivan said.

Soon after, the family was rescued by a Coast Guard helicopter and flown to an area hospital.

Sullivan’s brother, Christopher, told WCVB-TV that his mother was in critical but stable condition.

Naushon Island, part of the Elizabeth Islands, is owned by the Forbes family but has several coves that are open to the public, according to a company that offers tourist excursions to the islands.