MARATHON, Fla. — Eight relatives had set out to fish in less-than-ideal conditions off the Florida Keys. It was raining, seas topped 7 feet and winds were whipping up to 38 mph.

Before they knew it, two waves hit, capsizing their anchored 22-foot boat and knocking them into the sea about 3.5 miles offshore Saturday. Seven, including a 4-year-old, survived by clinging to their capsized vessel and a small blue cooler for nearly 20 hours, suffering exhaustion, jellyfish stings and hypothermia. A 79-year-old woman, the matriarch of the group, was missing and presumed drowned.

“When the will to live kicks in, human beings can do amazing things,” Coast Guard Petty Officer Nick Ameen said.

The women grabbed the girl and the 2 ½-foot cooler. One of the men tried to rescue his mother, but she slipped through his grasp and disappeared into the water. The women said the boat turned over so quickly that there wasn’t time to grab life jackets for anyone except the child, said Kendra Graves, a seaman with the Coast Guard.

Almost immediately, the two groups — the three women and girl and three men — drifted apart.

Nearly a day later, they were rescued when a commercial fisherman spotted the men Sunday morning and alerted the Coast Guard, which found the women and the blue cooler several miles away in the warm waters. Those rescued were taken to a hospital with non-life-threatening injuries.

Florida law requires children 6 or under on a boat 26 feet or less to wear a life jacket if the boat is moving. If the craft is anchored or docked, they don’t have to wear one.

As the weather improved Sunday, fishing boat captain David Jensen headed out with customers to catch live bait. Off in the distance, he saw a large object floating in the water.

As he turned the boat to get closer look, he saw a man waving. At first, he said, he thought there was only one person holding on to the sunken boat, its bow protruding just a few feet out of the water. When he got closer, he realized there were three men.

“I tried to get them to swim to the boat, but they said they didn’t know how to swim,” Jensen said. “Then I had the mate throw them life jackets. One guy put on the life jacket and swam to the boat. The other two guys wouldn’t get off the boat.”

One of Jensen’s customers jumped in and swam over. He tied the boats together, and helped the other two men, one at a time, back to Jensen’s boat.

“They were exhausted. One guy overnight had lost his mother,” Jensen said. “He was very visibly upset, which was a little tough because he was the one who spoke the best English.”

Zaida San Jurjo Gonzalez died. Her son, Jorge Alejo Gonzalez, survived along with his wife, Tomasa Torres, the elderly woman’s daughter, Elena G. Gonzalez, and her boyfriend, Juglar Riveras.

Also rescued were Jorge and Elena Gonzalez’s uncle, Jose Miguel De Armas, his wife, Yunisleidy Lima Tejada, and their 4-year-old daughter, Fabiana De Armas Lima. All are from South Florida. The other survivors’ ages ranged from 30 to 62.