I know personally my body has (seemingly) felt fear independent of my mind, and I have heard other drivers talk about this too. Typically this happens when you (think you) know you can execute a corner flat out, yet your foot lifts off the throttle just a bit all by itself; is one example that springs to mind.
(Personal experience, and I'm shit, so pinch of salt.)
The difference with what you describe is that you have a time pressure - you're being passively carried into the state where you're afraid and reactions etc come in real time.
Specifically when doing free solo stuff - I have time to "override" my responses if that makes sense? It takes conscious effort but I am moving at my own pace, I can rest, take a deep breath, etc.
Repetition and training, along with conscious intentional recognition and resistance to the automatic response, I imagine. That's how practically any degree of emotional control is always developed.
I wonder how climbers overcome this?