Flight Instructor with student reported a near miss in the traffic pattern with an aircraft that failed to give a position report to the tower controller. The airport reportedly has no radar or ADSB capability.

Narrative

I was doing touch and go practice with a student pilot at ZZZ. We were number 2 in the pattern for that lap. We report company traffic in sight; and we are cleared number 2 for the touch and go. As we turn base; I make note of an aircraft on a 3-4 mile final; but never got any warning from tower that someone was out there. As we are on final short; I see on our ADSB-in the aircraft on final rapidly catching up to us. After we touch down; I hear tower begin to clear someone for takeoff; then cancel their clearance. Tower then asks the aircraft on final to execute their missed approach; and if they have us in sight. As this is happening; I take controls from my student. and immediately sidestep the runway and level off at 200 MSL; just above the tree level. The offending traffic made no evasive maneuver; and just continued on runway heading without climbing as far as I could tell. According to our ADSB-in the traffic was within 100 feet vertically with no horizontal separation. I turn a very low crosswind; and then bring us back into a normal downwind for a full stop landing. After landing; I asked the ground controller what happened; and he stated that the traffic failed to report the final approach fix for the RNAV. I believe that if the tower was allowed to use ADSB or radar for situational awareness; this could have very easily been prevented.