GA pilot reported a NMAC in the traffic pattern at a non-towered airport during departure with an aircraft on approach for the opposite direction runway. Pilot took evasive action to avoid a collision.

Narrative

The preferred runway is Runway XX at ZZZ when winds are calm. I had to wait several minutes to take off on Runway XX while a C172 and a Pipistrel Alpha landed on XX. Each aircraft made all the appropriate radio calls. I made a backtaxi call and a takeoff call. On the departure leg I looked at the ADSB traffic before turning crosswind. I saw Aircraft Y approximately 1 mile ahead; 200 feet above; and to my right. So for right traffic(correct for Runway XX at ZZZ) I would be turning into him while climbing to pattern altitude. So I pulled the throttle back and leveled off while trying to get the aircraft in sight. When I eventually saw it I turned crosswind; avoiding him; and told him he was flying into the wrong runway. He said he had been making radio calls and was going to continue straight in to XY. I asked if he heard my radio call taking off and he said no. He continued to land on Runway XY even though he now knew that other aircraft were using XX. The other two aircraft ended up taxiing back to the FBO before he landed so there was no further conflict. I talked to the pilots of the 172 and the Pipistrel after landing and they confirmed that he was not making radio calls.