More than 100 people were thought to have been at the Cold Springs Swimming Hole in Tonto National Forest when the flood struck on Saturday afternoon.
Water Wheel Fire and Medical District Fire Chief Ron Sattelmaier said the flood was sparked by a severe thunderstorm in a remote area that had been burned by a recent wildfire.
About a dozen other people are missing.
Four people were rescued by helicopter and taken to hospital with hypothermia.
Mr Sattelmaier said: “They had no warning – they heard a roar and it was on top of them.
“It’s pretty much recovery (now). We don’t believe there’s anybody left out there.”
Footage recorded by Disa Alexander shortly after the flood showed a man in a tree holding his baby as water rushed around him.
The man’s partner was nearby, also clinging to a tree.
The flooding happened during monsoon season but it had not been raining where the swimmers were.
Mr Sattelmaier said: “I wish there was a way (of) keeping people from getting in there during monsoon season.
“It happens every year.”