Instructor pilot reported an opposite landing non communicating aircraft at the non-towered airport the Instructor and Trainee were using for practice landings. The other pilot told the Instructor he had fuel problems and also none of his aircraft's radios worked. The Instructor was unable to find registration information the conflicting aircraft.

Narrative

On DATE; we (instructor + student) departed ZZZ at XA:04 am for a local flight to ZZZ1. In flight; we listened to the AWOS at ZZZ1 and heard the winds were coming from 110 at 3 or 4 knots. ZZZ1 has Runway XY and Runway XX. Based on the wind direction; it was clear to us we were going to be landing Runway XX. Nobody was in the traffic pattern. We made a 5 mile southeast of ZZZ inbound Runway XX position report; 1 mile 45 right downwind Runway XX report; right downwind report; right base report; and final report. We touched down on Runway XX at XA:21 am and after being ~1;500' down the runway; we noticed a plane had just landed on the opposite runway and was a few thousand feet in front of us. He landed on the opposite runway; Runway XY. Runway XY was not the runway in use. I (instructor) stated 'my flight controls' and swerved us to the right off the runway. The other plane exited on taxiway D; hopped out of his plane; walked across the runway and approached us. I quickly jotted down his tail number; Aircraft X. It looked like it may have been a crop duster. He stated he had fuel pressure failure and his radios were not working. It appears Aircraft X isn't even registered with the FAA.