Flight Instructor reported a ground conflict with an opposite direction landing aircraft that resulted in an intentional runway excursion by the reporter.

Narrative

I was on a routine training flight with a newer Pilot in his taylorcraft bc12d. A simple vintage tail dragger with no electrical system. we were practicing landings at a small uncontrolled airpark called ZZZ. On this particular landing we had landed on the longer asphalt Runway rather than the grass Runway. This particular aircraft does not have ADSB and it has a handheld radio connected to an intercom. We had had intercom issues on departure but got it sorted out and used the radio extensively in the practice area on the way to the airport with confirmed radio communication no problem. Upon entering the area of the airport we switched to CTAF. There was no other traffic; but the frequency is shared with a bunch of small airports in the area so it should have been a clue that perhaps our radio was not functioning properly. I tried to call an airplane transiting the area earlier and got no response but thought maybe he was just passing through and thought little of it. anyways on this particular landing we were base to final for Runway XX and I see a 182 in a downwind for XY. This being an airport under a low bravo shelf; sometimes planes are fairly low and again I didnt think much about it. I made a 'yellow taylor craft on base for XX stop and go call' and even asked the 182 if he saw the taylorcraft. Again; no response but I thought it was similar to the other aircraft transiting the area. We landed with no issue and came to complete stop on XX with about half the Runway to go with an intention to depart for closed traffic. At this point; my student says the 182 is turning base to final and I see him about the same time. Honestly I froze; tried a couple of radio calls to no answer. I debated departing but thought that it would be a bad idea to be airborne with an aircraft short final opposite direction. Next thing I know he's short short final and my student says he's still on final. He has no intention of going around. Theres no nearby taxi way (locals often fly XY because the Taxiway is at the end of the Runway and it saves them having to back taxi) I went full deer and headlights. Never have I seen an aircraft not go around when theres something occupying the Runway...like an airplane; opposite direction; stopped. My Student is the one that recommended driving off the Runway into the grass. and we executed just that. We got kinda stuck in a gopher hole. I had to help the plane back onto the Runway; the 182 just went ahead and landed and rolled out to the end of the Runway like we were never there. My Student says it looked like the Pilot didn't see us until after he passed us but who's to know. After that near miss we departed without incident. I was pretty sure the radio wasn't working right. as we re entered the practice area heading towards ZZZ1 some jiggling cajoled it back into receiving. Upon entering the ZZZ1 traffic pattern it was working just fine again. A radio check confirmed that and everyone heard us just fine and we could hear everyone too. Should have been more aware that a quiet radio in a busy airspace is a funny place to be. Lured into a sense of security by a transiting aircraft and no radio response. No way I thought an aircraft would commit to land with an aircraft on the Runway opposite direction; stopped; even with no radio calls or radio call acknowledgement we were an obvious obstacle halfway down a narrow 40 ft Runway with nowhere to exit. Its a '47 with no electrical system; even if we didn't have a radio we would have been legal but the rule books are made of paper I guess. Prompt action by my student and a forced Runway excursion seemed to save the day. But seeing a 182 crossing the threshold with us staring at him was very very shocking... Everyone is trying to kill you I guess radio call or not. Sorry to be in your way on your way back to the hangar but it seemed like a go around would have been prudent.