A flight school instructor reported an aircraft landed behind them before they had fully exited the runway.

Narrative

I joined the left downwind leg for runway XX at ZZZ. At this time there were several aircraft in the pattern. Despite this; we were able to turn left base at a normal distance from the runway and execute a touch and go without issue. On the following lap of the pattern; we were number 3 to land; and the downwind leg was stretched 2nm beyond the threshold. As we turned left base; I noted an ADSB track depicting an aircraft joining final from right base with a higher rate of speed. At no point did this aircraft make any position reports. We continued our approach and landing and exited at the first available taxiway. As we turned right to exit the runway both my student and I looked back and to the right to observe our separation from the trailing aircraft. The SR20 (Aircraft Y) passed behind us in the landing attitude with his wheels just touching the ground. Our tail had just cleared the runway and no part of aircraft had passed the runway threshold markings. I estimate that the minimum separation was under 50 feet. Had Aircraft Y made position reports I may have been more aware of our lack of separation and been able to take corrective action in order to avoid this near miss. Had Aircraft Y made an AIM recommended pattern entry he may have been able to better maintain a safe distance from our aircraft and account for the significant closure rate on final approach. Aircraft Y should have executed a go-around once the lack of sufficient separation became clear. Lastly; the presence of a control tower at ZZZ airport would; in my opinion; greatly mitigate the risk of runway incursions and midair collisions.