Single Engine non-flying pilot reported an airborne conflict while on final approach at a non-towered airport. The conflicting aircraft announced making an approach on the opposing end of the current runway against traffic and current wind conditions. Reporter made a missed approach and the other aircraft left the traffic pattern while still not responding to any radio calls.

Narrative

IFR flight plan to ZZZ; did the RNAV approach from ZZZZZ (IAF) cleared to land on Runway X (winds were favoring Runway X); canceled IFR with ZZZ; switched over to local UNICOM; there was an aircraft - Aircraft Y- in the pattern announcing Runway XX. Radioed Aircraft Y for them to check the winds and that Runway X was the better runway - that we were on final Runway X for a full stop landing. Aircraft Y continued to radio that they were turning to final for Runway XX. We radioed back that we were on final for Runway X and that Runway XX was not the appropriate runway for the winds. When we were on short final Aircraft Y called that they were 'going missed or going around' and we radioed back for them NOT to climb because we were directly above them; at this point we took action and leveled off; watched as Aircraft Y turn left and climb alongside of us while we overflew Runway X. We radioed them to call us and when we received no reply said we would make a report. Aircraft Y then departed the area. We landed Runway X afterwards. Although Aircraft Y was making radio calls; they never responded to us. I can only assume Aircraft Y was practicing approaches VFR on Runway XX because Runway X approach is in the class Bravo airspace.