General aviation instructor pilot reported a near miss at a non-towered airport when on the landing roll was overtaken by the other aircraft on a go-around passing just above the instructors aircraft. The instructor performed a full stop and then watch the same aircraft land behind another aircraft who was still on the runway.

Narrative

Person A (CFI) and Person B (Student Pilot) were conducting a crosswinds landing lesson at TPL Temple Texas in a Aircraft X (Six Pack airplane; ADSB-out; no TCAS; only instructor provided Sentry ADSB-In Device); when a Aircraft Y; Piloted by Person C (Student Pilot) following us in the traffic pattern; as he approached behind us; Approx. 1.5 NM behind but overtaking in speed; according to my sentry device; I hailed the Pilot on CTAF to the effect of Aircraft Y be advised we are 58 kts Ground Speed. 1st time transmission we received no response; 2nd attempt I asked if he had us in sight and informed of our speed once more; as he reached a mile behind according to the sentry and still overtaking in speed. He responded saying he has the traffic and will go around if needed. As Aircraft Y got closer; he announced he was initiating a go-around; but his further actions dictated otherwise; and he was approx. 0.3 NM away at this time and above us according to the sentry. We were already established on and lower on the final approach; and we had right of way as per the FARs. We were #1 to the runway in use and on short final; over the approach lights at this point; and understanding that the following Pilot was going around; we continued for the landing to which was announced as a touch and go but at this time we changed to say it would be a full stop due to the evolving situation to get on the ground and out of the way and to decompress and talk about what just occurred. At this point I; Person A; was already on the controls and in command of Aircraft X over the runway and in the flare; now we get the alert he is +01 (100 feet) above and 0.1 NM away and I asked if he was going around; while realizing he is right over top of us via looking outside and seeing a large plane shaped shadow and then followed by seeing him just above; approx. 50-100 feet above and to the left of us; leveled off and not in a climb/go-around. Going around was not a viable option for us as we could not see him well due to us; the high wing on the lower approach; and him in a low wing on the higher approach and behind us; and with the understanding that he was executing the go around as per his last call on CTAF; our best and only option was to continue and land. In this moment; I thought to myself for safety of my student and myself; I would rather be hit on the ground than in the air. As we got clear of conflict finally; we taxied back to the active Runway 16 at TPL and held short. My student began to record as well as I did when I determined it was safe to do so holding short and in control of the plane; for evidence of the next incident/near miss. This was a near miss incident that could have resulted in the loss of life due to student pilot negligence while on a solo flight as determined by other radio calls exchanged between the 2 primary involved aircraft; and could have been avoided by using common tactics to buy time and distance; e.g. extending the downwind or going around and side stepping off the runway and making these calls on the CTAF or just better speed and distance management.2nd Incident/Near miss; ObservedAfter what had just ensued; Aircraft Y climbed away and entered back into the pattern he circled back around for another approach/landing attempt; we at this time are holding short of 16 @ A Taxiway and are in full view of Aircraft Y next approach for landing where he again is approaching fast and close behind Aircraft Z; Aircraft Y; continuing to land less than approximately 1000-2000 feet behind Aircraft Z while they are still on the active runway and most likely unbeknownst to them of how close the following traffic really was. This was a near miss incident; and presumably a runway incursion; that could have resulted in the serious injury and/or damage or even loss of life due to student pilot negligence while on solo and could have been avoided by using common tactics as well to buy time and distance; e.g. extending the downwind or going around and side stepping off the runway and making these calls on the CTAF or just better speed and distance management.