Flight Instructor reported a near miss while on the landing rollout with another aircraft that attempted to land in the opposite direction without communication. The instructor maneuvered off the runway to avoid a collision; resulting in a flat main tire.

Narrative

We were approaching 63S (Coleville) on a left downwind entry on the 45 for runway 19 '. We max an initial call 6 or 7 miles. Winds were calm/ 230 @4 (checked again after landing) we made an additional call entering the left downwind for runway 19 and base to final. After touchdown I noticed a small aircraft on very short final about to touchdown runway 1. I immediately took controls from my student and applied max braking and exited off the side of the runway to avoid a possible collision. Right when I did that I noticed the small aircraft going around. We pulled off runways and notice a de beaded tire. The small aircraft did not make any radio calls. I made a radio check on CTAF and other pilots responded saying loud and clear. The small aircraft kept flying north after the go around not making any radio calls and eventually left the area. Finally 15-20 mins after the incident we heard the first radio call from the aircraft which was the other airplane making no calls. Pilot may have been having radio troubles. Everyone ended up safe no damage other than a flat tire.