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.
Officials said it was sparked by a severe thunderstorm about eight miles upstream which sent water rushing down to where the family swam.
Authorities and a family member identified the dead as two-year-old Erica Raya-Garcia, Emily Garnica, three, Mia Garnica, five, Danial Garnica, seven, and Jonathan Leon, 13.
Also killed were Javier Raya-Garcia, 19, Celia Garcia Castaneda, 60, Maribel Raya-Garcia, 24, and 26-year-old Maria Raya-Garcia, who was celebrating her birthday.
Maria’s husband, Hector Garnica, is still missing – their three children are among the dead.
Hector Garnica’s parents were dining at a restaurant where their 27-year-old son cooked when they received the news.
They had been awaiting his return from the swimming hole where he was celebrating his wife’s birthday in the company of three generations of their family.
The manager of the restaurant where Hector worked said: “They’re like the Brady Bunch, they’re just extremely close.
“It’s pretty impressive how close they all are.
“I have nothing bad to say about him, you won’t find anyone in this town that has anything bad to say about the guy… everybody has great things to say about him.”
“An entire young family was wiped out,” said Detective David Hornung, of the Gila County Sheriff’s Office.
“These people are devastated, you cannot describe how they feel.”
He said the search was ongoing for the missing father and the hope was to find him alive.
“I’ve seen miracles happen before,” Hornung said.