C-172 Instructor Pilot reported an aircraft stopped and blocked their aircraft from exiting the runway. Another aircraft elected to commence a takeoff run with the Instructor Pilot's aircraft unable to clear the runway.

Narrative

On the morning of day 0; I was conducting cross country/insurance required flight training with a certificated private pilot. After landing Runway XX ZZZ1; the first turn off/taxi way was blocked due to Aircraft Y; (see additional details below). Consequently; we planned to continue to the next turn off. However; as we approach the first turn off Aircraft Z; announced that they were departing Runway XX. As we looked behind us down the runway; we saw the aircraft quickly approaching us. We immediately took evasive action and departed the center line towards the blocked turn off/taxiway. Within seconds; Aircraft Z passed behind us and continued their climb out. My observation of the aircraft blocking the first turn off/taxiway; (Aircraft Y); was that they did not appear to be aware of the situation behind them. In fact; after the aircraft passed behind us; a person got out of the right seat of Aircraft Y and walked down the taxiway to the flight school. After Aircraft Z announced that they were departing Runway XX and started rolling; I made a radio call stating that we were still attempting to clear Runway XX. No aircraft acknowledged. All of our anti-collision lights were illuminated. Aircraft Z subsequently departed the pattern without incident.