Instructor pilot reported a near miss during a training flight in day VMC conditions at a non-tower controlled airport; while in the traffic pattern. The instructor took the flight controls and maneuvered away from an aircraft entering the pattern; and then performed another evasive turn and descent to avoid a collision with another aircraft; then landed safely after the maneuver.

Narrative

I was providing flight instruction to a student pilot conducting traffic pattern work at the MKY airport using Runway 35. We had been there for four (4) touch-and-go's on our fifth (5th) lap extended our downwind due to other traffic that was ahead of us in the pattern. While we were on short final the aircraft that caused the near midair collision (Aircraft Y) made a call on the CTAF that we had delayed them by two (2) minutes. After our fifth (5th) touch-and-go Aircraft Y announced they had taken Runway 35 and were about to depart. During our upwind leg I noticed a different airplane (Aircraft Z) was southbound heading directly towards us and descending to pattern altitude. Because of this; I took the controls from my student; started our crosswind turn at 700 ft. and announced on the CTAF that we were turning crosswind and that we were remaining in the pattern. I did this to clarify to all other traffic that we were not departing and would be climbing to and remaining at pattern altitude during the crosswind and downwind legs. As we started turning into the downwind I announced our turn and again repeated that we would be remaining in the pattern. At 900 ft. I looked to my right during the turn to downwind and saw Aircraft Y extremely close to us and approaching at the same altitude so I took evasive action to avoid a collision. I immediately cut the power to idle and initiated a rapid descent to the right because the other aircraft (Aircraft Y) was climbing and turning to their right. At no point did Aircraft Y announce that they were going to start their crosswind turn; nor did they announce that their intentions were to depart to the East. Had they done so it would have indicated to me their direction of flight and allowed myself and my student adequate time to avoid their flight path. Seconds after the near collision another pilot on frequency said 'what [expletive] were you doing you almost hit that Aircraft X.' Another pilot at MKY said they were watching from the ground and it was 'scary to watch'. Proper communication and observance of the guidance set forth in the FAR/AIM and advisory circular 90-66C would have mitigated the confusion and lapse of safety caused by Aircraft Y. It is clear that because we delayed Aircraft Y by '2 minutes' the pilot felt the need to disregard safety and jeopardize lives.