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The 41-year-old mother-of-five left the Channel 4 reality show just minutes into the second episode, when the recruits were woken up after just one hour of sleep and told to stand holding a heavy sandbag over their head.
Price, who had already been labelled by ex-forces instructors as "the weakest link", struggled to lift the bag and after resting it on her head, gave up and threw her number onto the ground in front of officers as she stormed off camera.
She admitted afterwards: “I’m so p**sed off. I didn’t have the strength. I will always f**king beat myself up. So p**sed off with myself.”
It had previously been reported that Price – who was declared bankrupt last year – had quit the reality show after two days of filming.
In an interview filmed before her departure she said: “It doesn’t matter what anyone says to me I can cope with it. I’ve proved it but I won’t be the fittest and I will struggle.
“It’s important for me to stay strong. I feel I’ll come out a new person if I push myself. It’s a tough thing to do this course, it’s tough.”
She has since claimed she left the show on medical grounds, saying: “I pulled something in training. My breast reduction became inflamed. It was out of my control. I was gutted. I had to leave because of the pain I was in.
"I’ll never take my creature comforts for granted again. I left with new life goals. I left the old Katie up there. She’s not coming back.”
Blue Peter presenter Anthea Turner, 59, also quit the show later in the second episode, after being the last to return from a gruelling hike.
Other contestants taking part in the new series include Strictly Come Dancing’s Brendan Cole, TV personality Joey Essex, Blue Peter presenter Helen Skelton, former Premiership footballer John Fashanu, Hollyoaks actress Nikki Sanderson, YouTuber Jack Maynard, Paralympic athlete Lauren Steadman, Rudimental DJ Locksmith, Radio 1Xtra DJ Yasmin Evans and former boxing champion Tony Bellew.
Celebrity SAS: Who Dares Wins continues on Channel 4 on Mondays at 9pm.