Small Aircraft; Low Wing; 1 Eng; Fixed Gear
Flight Crew Flight Instructor; Flight Crew Instrument; Flight Crew Commercial
Workload; Time Pressure; Communication Breakdown; Distraction; Other / Unknown; Situational Awareness
Conflict NMAC
Flight Crew Took Evasive Action
Human Factors
Shot the RNAV XX at ZZZ and flew published missed. As we approached the hold at ZZZZZ; noticed non company aircraft on ADS-B +800 ft. and approximately 7 miles left of us on an intercept course. We radioed our missed approach and monitored frequency. As we approached the hold entry at ZZZZZ; non company aircraft began descending toward us (we were visual and they were low wing). In anticipation we started descending to try to get under them. They began turning left  approximately 500 ft. above us towards our path. At this point we lost visual with them and I took controls to turn to the right to avoid them as they descended lower. They continued their left turn and we regained visual with them at the same altitude. They continued descending as we flew opposite. Probably came within 400-200 ft. Could have monitored other frequencies to establish radio contact.
Flight Instructor reported a NMAC while attempting to fly the published missed approach from an  RNAV approach. The other aircraft was descending through the Instructor's altitude and there was no communication between the two aircraft.