General aviation instructor pilot reported a ground conflict with another aircraft who did not properly space their approach at the non-towered airport and landed while the instructor was still on the runway.

Narrative

I was with a student practicing pattern work at AXH; we had just turned left downwind for runway 9 when two new airplanes [Aircraft Y and Aircraft Z] in quick succession radioed CTAF that they were to the east planning on entering the left downwind for 9 as well. The first [Aircraft Y] reported they were 3 miles east and the second [Aircraft Z] reported 5 miles east. I got on the CTAF again; in case they hadn't heard that we were already in the pattern; and reported that we were midfield on the left downwind. As we turned left base the furthest out AC (aircraft) [Aircraft Z] asked the closer AC Aircraft Y] what his position was and the closer AC reported he was in the left downwind abeam the numbers. We again called that we had just turned left base and were planning a full stop expecting the closer AC to extend his downwind for spacing. He did not. Instead he followed us at about half a mile base to final calling final on the CTAF right after us. Again; I got on CTAF and stated that we were short final and emphasized that we were a full stop. He landed on runway 9 about 1/3 a mile behind us when we were still very much on the runway. He then attempted to exit the same taxiway we had used to exit and continued to block the runway for several seconds while we cleaned up the airplane and executed our after landing checklist. The further AC extended his downwind as one would expect and landed without issue.