Instructor pilot reported an NMAC with a fast-approaching aircraft in the vicinity of MTV airport that was disregarding aviation right of way rules. The instructor used an evasive maneuver to escape a collision.

Narrative

During my training flight I was provided VTF for the RNAV 31 into MTV from the SW. I had my student descend to 2;600 ft MSL on a heading of 043. Another aircraft was doing some maneuvers in the area from previous tracks I witnessed on ADSB (Automatic Dependent Surveillance-Broadcast). I was positioned between [the other aircraft] and SIF; and under the impression he was doing maneuvers/transit flying. I notice on my ADSB he is now +500ft and heading in our direction. I am positioned to his right and at a lower altitude (providing me right of way) however for safety I had my student descend another 100-200ft. He was closing the gap rapidly horizontally with his speed and then starts descending on ADSB. At this point we were too close for me to trust ADSB direction and altitude readings (due to refresh times). I took flight controls and began a rapid descent (lower altitude aircraft descends) to avoid the conflict. As I am making this rapid descent he was almost matching my descent. He NEVER made any correction We were tuned to 122.7 (MTV CTAF) to listen for traffic flows; however at this point I considered he was doing a practice approach in SFI. I quickly switched over to 122.8 (SIF CTAF) as I switched I heard the pilot say 'I think I'm going to hit you'. At this time he was STILL descending into me made no deviations at all. This also confirmed he clearly saw me on ADSB and never cared to make corrections. I kept my rapid descent to avoid any collision and he continued his practice approach; seemed like he was doing RNAV 13 circling 31 into SIF. I tried to slip to increase descent and raise my wing to get a visual. I never got a visual; and at time of passage ADSB reported +100. I tried to maintain separation and get to the radios; but there was no time for how fast he was approaching. We were safe and clear; so I continued my operations. I believe he recognized where I was and believe he had priority doing a VFR practice approach; disregarding right of way rules and the fact I as descending to get out of conflict. Again he never made any attempts to deviate the conflict. The airports are in close proximity; so are the approaches.