An Instructor Pilot reported a NMAC approaching the traffic pattern at a non-towered airport.

Narrative

Flying back from ZZZ to ZZZ1. As we start approaching ZZZ1 we hear a king air aircraft announce that they are over flying the field at 5900. We see them on ADS-B. At this point we are about 10 NM away from ZZZ1. We see the King Air turn towards the north. We announce that we will stay to the west and asked that the King Air stays to the east. We got no response. We decide to deviate towards the east since the King Air is showing now intentions of deviating towards their right; which would be towards the east. As we start to turn east the King Air suddenly turned towards their right; so we immediately deviate to our right. We attempt again to make contact with the King Air but they do not respond. We get visual contact and deviate around them. Immediately after passing them the King Air turns towards us and starts flying directly towards us. This put the King Air at our 5-6 o'clock position. At this point I take controls and immediately begin a climb; completely ready to bust the Class Bravo at 6500 ft msl should we need to. For the third time I attempt to make contact with the King Air by tail number but again they did not respond. When we were close enough to be able to see the pilots of the King Air; approximately 0.1-0.2 nm away;they saw us and did a 360 degree turn away from us. We land at ZZZ1 without further incidents. The ultimate cause of the near miss was a King Air maneuvering under a Class Bravo shelf 2.5 nm wide and only about 1800 ft tall and the King Air pilot not being on frequency in a very busy training airport.