PA-34-200T Turbo Seneca II
Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 120; Flight Crew Total 2100; Flight Crew Type 300
Communication Breakdown; Training / Qualification; Situational Awareness
Conflict NMAC
Flight Crew Took Evasive Action
This was a practice approach to ZZZ RNAV XX.  I was the Instructor in the right seat.  We had requested the practice approach from ZZZ Departure and was cleared for the approach and had changed to the ZZZ CTAF frequency. We had made several radio calls of our location. Our airplane has a Garmin 530W with ADSB and Traffic. Also onboard was an iPad with Foreflight and a Status 3. At no time did we get any traffic warning or observe Aircraft Y in the pattern at ZZZ. We were on short final at about 1/2 SM and 500 MSL. When my Pilot (Student) spoke out of the near aircraft; as I looked out the right window I observed Aircraft Y within 100 feet.  I immediately took the flight controls and made a descending right turn towards the south away from the aircraft.  I then transmitted that we were turning to avoid the aircraft that had just flown over us.  There was a long pause when a female voice transmitted that they didn't see us.  We circled around and re-entered the pattern to land.  The other aircraft had continued their approach and landed.  After we landed and taxied around for another takeoff; I observed the plane parked at the fuel with two individuals standing near the aircraft doors. When the aircraft departed; our equipment showed the aircraft but no display of a tail number.  Prior to this incident; our equipment had been reporting other traffic.
PA-34 flight instructor reported an NMAC while on approach to a non-towered airport.