Saturday's 3-0 home defeat to Crystal Palace left City six points from safety and on a run of just one win in nine league games.
Sky Bet now make Ole Gunnar Solskjaer's side 1/11 favourites to return to the Championship after just one season, but captain Caulker is not ready to give up.
He said: "To be honest, I just think we lacked that cutting edge against Palace. That little bit of fight too.
"We apologise to the fans that turned up and gave us great support, as always.
"We've given ourselves a mountain to climb now, so we are going to have to dig deep and show a lot more fight and passion than we did against Palace. It's going to be five cup finals between now and the end of the season.
"It's still mathematically possible we can stay up, so we are not going to write ourselves off.
"We've got confidence from how we played at the start of the season. We've got to get back into some kind of form, but every game now will be a fight."
Solskjaer struck a similarly defiant tone in the aftermath of Saturday's defeat, saying: "Psychologically, it is a proper kick in the teeth.
"But I am not just saying it - I have got lots of fight, I have got loads of belief. I know the players, and they deserve to hear that I still believe they can do it because I see them every single day. Most of these are fantastic.
"Mentally, it has been a very draining season for many - the first time in the top flight in 51 years. It showed in the stands as well.
"The players heads dropped, we ran out of ideas. Of course, that is difficult. It is going to be hard to pick them up, but we have to find a way. That is just the way it is - you have to find a way. Can we produce a miracle? That is what we have to do."