Flight Instructor reported an aircraft landed on the runway behind them while they were still on the runway back taxiing.

Narrative

After returning from a routine training flight; my student and I were established in the downwind. While clearing base; I noticed oncoming traffic for what appeared to be an extended final Runway 10; same runway we were landing. I established radio communications with the other aircraft at this point; they previously made no radio calls. He reported having me in sight only after I inquired. After resolving the issue and assuring proper separation; my aircraft turned base from an extended downwind. At this time; the other pilot made a descending left turn from 1800 ft. to a high base; all 1/2 mile west of the approach end of Runway 10. He was again pointing toward me as I was now on final. I repeatedly informed the aircraft of our intent to land and back taxi; as our parallel taxiway is closed per NOTAM. My student landed while I monitored ADS-B In on my EFB. This was our most valuable tool and gave increased SA (Situational Awareness) which resolved what could have been several collision hazards in the pattern before they even became an issue. While on short final; I again called CTAF explaining my intent to land and back taxi. My EFB showed him 300 ft. above me and 1/4 mile behind at this point. Expecting him to go around; I did not execute a go around as I was lower in the pattern and on short final. After landing and turning to back taxi; I saw the offending aircraft sitting on the Runway 10 numbers. He had landed seconds after me; behind me by about 800 ft.; on an active runway. Radio comms directed to his specific tail number increased SA and prevented any collisions. But his omission of radio calls; SA; non-standard pattern use; and dangerous ADM caused a runway incursion.