Three sisters who were helping celebrate their father's birthday are thought to be among the 34 people killed when a diving boat caught fire off the coast of Southern California.

Susana Rosas of Stockton, California, posted on social media that her three daughters, their father and stepmother were on board.

“It is with a broken heart ... 3 of our daughters were on this boat. As of now they are still missing,” she wrote. "My #1, Evanmichel Solano Quitasol, my #3, NicoleStorm Quitasol and my #4, Angela Rose Quitasol.

“The authorities do not have much to say to us." She added: “Thank you to all of you for your support prayers and good wishes.”

It is believed the three sisters were on the boat to celebrate the birthday of their father, Michel Storm Quitasol, along with their stepmother, Fernisa Sison.

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Brett Harmeling thanked people for their “unconditional love and support during this incredibly tragic time” in a Facebook post after it was reported his sister, 41-year-old marine biologist Kristy Finstad, was aboard the Conception when the fire erupted.

“No final word on my sister Kristy; however, it is likely she has transitioned to be with the good Lord,” he wrote.

Authorities have not released any victims’ names, but Santa Barbara County Sheriff Bill Brown said he had heard anecdotally that they ranged from teenagers to people in their 60s.

He said most appear to have been from Northern California, including Santa Cruz, San Jose and the San Francisco Bay Area.

Evita Olson is helped by her two sisters, both presumed to be victims of the California boat fire, as she dresses for her wedding in the photo on the left. On the right, Kristy Finstad, another presumed victim, shares a photo from her travels as a respected diving instructor around the world. (Facebook)

Santa Barbara County Fire Chief Mark Hartwig told reporters local officials were “sensitive to the fact that families have gathered today, some from outside of the area, to bring their loved ones home”.

Those families “will rely on us to do everything in our power to find out happened aboard that vessel in the last moments of these family members’ lives”, he said, adding: “That’s our commitment.”

Pacific Collegiate School in Santa Cruz, an elite California charter school, said some of its students and their parents were on the boat.

The school recorded a message on its answering machine that said: “Our hearts and prayers are with the victims and for the families of those missing, particularly those of our students.”

The Coast Guard called off its rescue operation on Tuesday after determining there were no more survivors, Captain Monica Rochester said.

DNA will be needed to identify all the victims, and authorities will be using the same rapid analysis tool that identified victims of the deadly wildfire that devastated the Northern California town of Paradise last year, the sheriff said.

It is not yet known what caused the fire early on Monday. The sheriff said the fire spread rapidly and flames blocked both exits out of the lower deck, where passengers and one crew member were sleeping in tight quarters.

The captain and four crew members awake on the upper decks jumped off the front of the vessel, swam to an inflatable boat at the back and steered it to a ship anchored nearby. Authorities have interviewed them but haven’t said what efforts they made to help the 34 people trapped aboard before abandoning ship. The crew member presumed dead was asleep below deck with the passengers, Mr Brown said.

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It appears the crew sent multiple mayday calls, including one from the Conception in which a crew member could be heard saying he couldn’t breathe.

Twenty bodies have been pulled from the sunken vessel that had been anchored close to the shore of Santa Cruz Island, about 20 miles (32 kilometres) off the coast and northwest of Los Angeles. Four to six other bodies have been spotted underwater, and divers are looking for the remaining people who are missing.