Instructor pilot reported taking evasive action in a critical ground conflict situation at a non-towered airport.

Narrative

Summary- practicing in the traffic pattern at non-towered airport. An aircraft (Aircraft Y) enters the area and conducts a visual approach on the opposing runway (XX) without making any calls of his intentions. As I landed on Runway XY; I noticed Aircraft Y descending on a short final. I made the corrective action to abort takeoff while Aircraft Y executed a missed approach; continued in the pattern; and landed uneventfully on Runway XY. My student and I were practicing traffic patterns at ZZZ. There is only one runway located at ZZZ; Runway XY and XX. Around XA:10-XA:15; I noticed my adsb-out alerting 2 incoming aircrafts arriving in the vicinity. To my knowledge; only the non-involved aircraft identified the appropriate runway. Aircraft Y did not make calls pertaining to which runway he was entering on but addressed that his approach was from the south. As my student and I were in the pattern; we were making our usual position reports (crosswind; downwind; base; final). On the base leg; we noticed one aircraft was already established on an entry for the downwind; while Aircraft Y (involved aircraft) was entering the area by showing a flight path towards the departure end of Runway XY. I did not perceive his entry at the time as an approach. I assumed he was maneuvering onto the left downwind for Runway XY behind the other aircraft. We continued our approach and made a radio call addressing our intentions of a touch-and-go. At touchdown; we intended to do a touch-and-go; unfortunately; towards the departure end of Runway XY; we saw Aircraft Y in a descent heading straight at us. I took action with my student to abort takeoff and made a radio call. We stopped before the end of the runway and cleared the runway. At 300-500 ft AGL; Aircraft Y added power and turned right to intercept the left downwind for Runway XY. My student and I taxied back to Runway XY and held short for both aircraft to land before taking off again. What I believe caused the runway incursion was the lack of situational awareness from the pilot or pilots of Aircraft Y. As a flight instructor from a flight school located at ZZZ1; we use this airport frequently to train students. It is known at my flight school that this airport has had many problems with general aviation pilots being unsafe. My intention for writing this report is to address the ongoing occurrences at this non-towered airport which many flight schools use as a learning area.