Flight Instructor reported a near miss in the traffic pattern at a non-towered airport. The instructor identified the aircraft on ADSB improperly entering the pattern and passing directly under the instructor's aircraft.

Narrative

At approximately XA30 PDT; Aircraft X was on an instructional flight at CXP. Winds at time of arrival supported use or Runway 27 which Aircraft X was using for pattern work. Aircraft X departed Runway 27 using standard pattern calls as well as establishing communication with other aircraft in and near the pattern when necessary. An additional aircraft (not included in this report) was also using Runway 27 behind Aircraft X and was also making standard radio calls. As Aircraft X climbed out; a fourth aircraft (not included in this report) and hoping to enter the pattern for Runway 9 and asked for a PIREP on the winds. Aircraft X reported that it had entered the downwind for Runway 27 and was climbing out and would leave the pattern to the east on the downwind for Runway 27. Shortly thereafter; Aircraft Y reported that it was 2 miles to the South to land Carson City (no runway given; also first call on the CTAF). The aircraft that had departed behind Aircraft X then reported they were number two on a downwind departure from Runway 27. Aircraft X was in the midfield; downwind position for Runway 27 approximately 1100 feet AGL. Instructor onboard Aircraft X noticed Aircraft Y approaching their position on ADSB traffic with negative visual contact. Aircraft Y passed directly under Aircraft X; 300 feet below according to ADSB data. Aircraft X never had visual contact with Aircraft Y until Aircraft Y passed underneath. Aircraft X also did not think it was appropriate to deviate heading since one aircraft was directly behind. Aircraft Y essentially entered a mid-field; downwind for Runway 9 (no 45 entry); dove under conflicting traffic; flew a right traffic pattern for Runway 9 below TPA while two aircraft were departing the left traffic pattern on the downwind for Runway 27 with those two aircraft making good position reports in the process.