They used helicopters and drones, arrived on horseback and in trucks, and searched from boats and golf carts. Sometimes, they muscled through brush and devastation on foot.

Emergency responders, family members, bystanders and others have saved hundreds of people from the devastation of the flooding in Central Texas, holding onto hope as national crews extend their search-and-rescue operations into a third day.

Many stories over the past 48 hours have been harrowing. A 22-year-old woman was rescued after clinging to a tree overnight. A young girl was found after floating on a mattress for hours. A mother and her 19-year-old son survived by clinging onto each other and a tree. Two young girls were found 27 feet off the ground in a tree near Comfort, Texas. A counselor at Camp Mystic helped evacuate her 14 young campers to safety.

One man, Scott Ruskan, a Coast Guard rescue swimmer, has been credited with saving 165 people. This was his first rescue operation, said Rachel Ault, a Coast Guard spokeswoman.

There were 12 rescue helicopters, including ones from the National Guard and the Army, on site at Camp Mystic, the Christian summer camp for girls on the Guadalupe River that was devastated by flooding. Mr. Ruskan “was the main person on the scene triaging everyone,” Ms. Ault said.

More than 850 people had been rescued as of Saturday night, and more than 1,700 people have been involved in the operations, officials said. Eleven girls from Camp Mystic remain unaccounted for.

“We will be relentless in going after and assuring that we locate every single person who has been a victim of this flooding event,” Gov. Greg Abbott of Texas said on Saturday at a news conference. “We are not going to stop today or tomorrow. We will stop when the job is completed.”

Mr. Abbott also requested a federal disaster declaration, a move that would direct federal assistance to the affected area. President Trump granted the declaration on Sunday morning.

Searches continued on Sunday even as forecasters warned of more rain and possible flash flooding.

City officials in Kerrville, the seat of Kerr County, have urged people not to go out to conduct search-and-rescue operations and have asked people not to use personal drones over affected areas.