Instructor pilot reported a near miss with another aircraft; during daylight VMC on a training flight at a non-towered airport. While climbing on a missed approach; the instructor observed an aircraft landing in the opposite direction; took control and banked sharply to avoid a collision; then resumed the flight.

Narrative

Prior to going the ZZZ asked us how will this approach terminate; we responded that we are going to do a touch and go and then execute the published missed. After turning on final; ATC told us to change to advisory frequency and that he will talk to us after we takeoff again. After doing the touch and go; we departed on runway heading; executing the published missed. Shortly after takeoff; around 1;400 MSL; I (CFI) heard the other aircraft make a radio call that they were on final for Runway 18 (which is the wrong runway at SLR); my student was flying and our pitched was up at Vy so our visibility was limited in front of us. After I heard this call; I pitched the aircraft down and saw the Aircraft Y just in front of us descending towards us. I immediately took the flight controls from my student and banked the aircraft to +50 degrees to the left to avoid impact. I inquired the other aircraft but they made no attempt to respond or make any more radio calls. I advised Center of the NMAC after establishing contact with them again flying the published missed.