Flight Instructor reported that just after liftoff at a non-towered airport another aircraft was landing on the same runway opposite direction. The instructor landed back on the runway but was not able to stop on runway remaining resulting in a runway excursion and aircraft damage.

Narrative

I was conducting a Commercial Pilot Multi-engine retraining flight with a student. The aircraft used for this flight was Aircraft X. The departure took place from ZZZ1; and the flight proceeded without any incidents reaching ZZZ2. As part of the retraining flight; a single-engine instrument approach circle-to-land was planned for ZZZ. Prior to entering the hold at the ZZZ VOR; weather conditions and relevant NOTAMs for ZZZ were obtained. The recorded wind at the airport was from 140 degrees at 10 knots; gusting up to 19 knots. The approach instructions for the ILS XX circle-to-XY approach were given to the student but with a modified MDA of 2;500ft MSL as to simulate the same app. Upon leaving ZZZ2; a radio call was made by me about our VOR inbound status and that approach altitudes were now open. On the final approach course; I conducted a simulated right engine failure for the student; using throttle control to simulate a single-engine approach. At approximately 5 miles from Runway XX; a radio call was made indicating our intentions for the 'option.' Around 2 miles from Runway XX; another radio call was made on the CTAF; stating that we were breaking off the ILS approach to circle for Runway XY at 2;500 feet MSL. A subsequent radio call was made while turning onto the base leg for Runway XY; informing that we would be performing a touch-and-go followed by a departure back to ZZZ2. As we were circling and approaching short final for Runway XY; the instructor of another flight school operated aircraft; Aircraft Y; made a 2-mile final approach call for a low approach to Runway XX. In response; an immediate radio call was made; clarifying that we were on final for Runway XY for a touch-and-go; there was no subsequent radio call from Aircraft Y. The student's landed the plane single engine; brought both throttles to close; retracted flaps and initiated his takeoff roll. On the roll; we obtained visual of Aircraft Y who was continuing for the opposing Runway XX and on a straight on course with our aircraft. With the student just initiating his rotation; I realized the approach aircraft appeared to be 100ft or less from the runway; I took controls of the aircraft to make evasive action. Over the CTAF; I advised the aircraft on final to go-around. With runway remaining; I brought the aircraft back to the ground and maximum braking was applied to bring the aircraft to a stop before the runway end. Despite our efforts; the airplane veered into the dirt at the approach end of Runway XX; resulting in a blown tire and damage to the left main gear strut. Fortunately; neither the student nor I sustained any injuries from the incident. There was no property damage to the airport incurred during this incident.