GA flight instructor reported a NMAC while on final approach. Flight abandoned approach and returned for landing.

Narrative

Today at around XA:00 at PIM me and my student were doing a landing lesson; everything was normal and there was about 3 of us all in the pattern at the same time; on our 3rd landing however; on our base to final turn we checked final to be clear and made the turn; we got established and that is when Aircraft Y came right over top of us and started to descend like it was landing; but its gear was up. Aircraft Y wasn't making any radio calls at all; even after I tried to reach him on frequency multiple times; me and me student decided to break off the final course and proceed north bound so that Aircraft Y wouldn't make the decent on us. After we turned Aircraft Y turned left at pattern altitude; broke off final; and started proceeding towards another plane in the downwind. At that moment it started climbing and flying towards LGC. On the ground I checked and saw that it was making a fuel stop at LGC. Assuming that Aircraft Y was on an IFR flight plan my assumption was that he was going direct to WAGUS; which is the IAF for the RNAV 31 into LGC and just so happens to be right over PIM; I am assuming that he got disoriented and mistook PIM to be LGC and started flying into it and was making calls on LGC CTAF; and that's why we couldn't hear him. However while I was flying I did not monitor LGC CTAF frequency. Why Aircraft Y was at pattern altitude for PIM at WAGUS doesn't make sense to me at all though; because WAGUS has a hard floor of 3000'.