General aviation instructor pilot reported a near miss during the initial climb after a touch and go; while on a training flight at a non-tower airport; in daylight visual conditions. An aircraft on final behind the instructor aircraft had inadequate spacing; performed a go-around then overtook the instructor aircraft; requiring the instruct to aggressively bank away to avoid a collision; then returned to the traffic pattern and land safely.

Narrative

Me and a student were practicing traffic pattern flying and landings in Aircraft X. Aircraft Y entered at a 45 with minimal radio calls. Prior to entry into the pattern they were 2;000 ft. above traffic pattern altitude. Therefore; they were descending into the traffic pattern. They cut directly in-front of Aircraft Z in the downwind giving very little spacing between us and Aircraft Z who was further behind. We continued with a normal approach and landing. After landing we reconfigured the aircraft and added power. Aircraft Y performed a go-around since we were still on the runway. They went around but held runway centerline. I asked them to offset from centerline but did not receive a reply. I grabbed the controls from my student and stopped the climb as I could see their shadow directly in front of us. I gained visual of Aircraft Y 300 feet above us and banked the aircraft away from them. I announced to them when they go-around they need to offset from runway centerline so aircraft don't climb into them. I never received a response. I believe the cause of the problem was the pilot being unfamiliar with non-towered airport operations and not offsetting when an airplane is climbing off the runway. This situation can be corrected if the Pilot in Command of Aircraft Y undergo training for non-towered operations and procedures.