A Flight Instructor reported a NMAC during initial climb from a non-towered airport.

Narrative

While upwind at ZZZ; SP (Student Pilot) was demonstrating a normal takeoff; slight crosswind was present. Around 500-600ft; SP pointed to traffic maybe 100-150ft above us cutting through the upwind/crosswind probably at or below TPA (Traffic Pattern Altitude). It was a Cessna. They had no ADS-B and made no calls on ZZZ [UNICOM] or ZZZ1 [UNICOM] freq. SP pitched down to avoid them and myself as well as the backseat pilot kept eyes on the Cessna as they departed over the runway toward Location X. We watched them climb and descend a bit over the mountains but eventually lost sight. We heard a call on our way back to ZZZ1 reporting their position over Location X requesting to land at ZZZ1 and they notified tower they had no ADS-B; which is why we suspect it could be them. It didn't make sense for them to be flying at or below TPA over a private field nor that low along the mountains especially with no ADS-B. That is also within the 30 NM B-ring for xpdr (Transponder) from ZZZ2; which seems unusual. We learned to always be eyes outside on every leg of the traffic pattern and not to rely solely on the MFD (Multi-function Flight Display) or iPad for traffic advisories in situations like this.