Can an extreme fight or flight response cause mental impairment?

Yes: Yes - the body's short-term response to the overwhelming flood of adrenaline can impair a reasoned response.

Possibly: Extreme fight or flight response equals extreme reaction to stress, which triggers the release of a harmful cascade of neurotransmitters and hormones, like cortisol, affecting negatively all body functions & organs, especially the brain. Acute stress disorder and post traumatic stress disorder are well established mental impairments directly related to exposure to high stress situation.