Can calmness happen during the fight-flight response? In the question How do certain individuals, like Quang Duc, develop the ability to remain calm when enduring significant nociceptive pain?, one answer says that it was the high level of cortisol that numbing his pain. But that's numb, not calm. I don't think calmness can happen during stressful situations.

So to check that answer, can calmness happens during the fight-flight response?
With looking at the possibility of calmness during stress and trauma, you only have to look at the situation with rape victims and soldiers witnessing severe traumatic events who dissociate (Ellert, et al. 2011)(Waller, et al. 2001) during the event.

Dissociation is a psychological defence mechanism (Cardeña, 1994) which helps the person to get through the situation with as little harm as possible.  Think about it like throwing a switch on conscious awareness.

Whilst some define dissociation as a combination of 2 distinct psychological mechanisms (Brown, 2006), the psychobiological mechanism of dissociation is little understood; but, when someone is dissociated from the event, the detachment of conscious awareness from the event can make the person appear very calm.

References

Brown, R. J. (2006). Different types of “dissociation” have different psychological mechanisms. Journal of Trauma &amp; Dissociation, 7(4), 7-28.DOI: 10.1300/J229v07n04_02

Cardeña, E. (1994) The domain of dissociation. In: Dissociation: Clinical and theoretical aspects, Edited by: Lynn, S. J. and Rhue, J. W. 5–31. New York, NY: Guildford Press.

Ellert R. S. Nijenhuis PhD &amp; Onno van der Hart PhD (2011) Dissociation in Trauma: A New Definition and Comparison with Previous Formulations, Journal of Trauma &amp; Dissociation, 12(4), 416-445.DOI: 10.1080/15299732.2011.570592

Waller, G., Hamilton, K., Elliott, P., Lewendon, J., Stopa, L., Waters, A., ... &amp; Hargreaves, I. (2001). Somatoform dissociation, psychological dissociation, and specific forms of trauma. Journal of Trauma &amp; Dissociation, 1(4), 81-98.DOI: 10.1300/J229v01n04_05