Flight Instructor with student reported taking evasive action to avoid a NMAC at a non-towered airport.

Narrative

This event took place during a flight today apart of a two hour stage 2 commercial stage check. Towards the end of our flight; we entered the traffic pattern at ZZZ airport. We remained in the pattern for approximately 25 minutes while completing 4 traffic lap patterns. After our final Lap; we departed Runway XX and experienced an immediate traffic avoidance situation. On upwind for runway XX at 1;700' MSL; 900' AGL; we noticed an aircraft off our right side coming from the northeast. Flight Instructor (IP) was at the flight controls while Student Pilot (SP) helped with traffic avoidance as previously briefed. IP maintained aircraft control and appropriate visual scan. SP noticed this aircraft off the right side just prior to IP completing scan to the right side. IP made immediate corrective action by pitching down to get below this incoming traffic. After making this corrective action; the danger was alleviated. The call sign off this incoming traffic was Aircraft Y. No radio calls were observed from Aircraft Y until they passed over us and entered the left downwind for runway XX. This radio call from Aircraft Y was observed after crossing over us on the departure end of runway XX. Aircraft Y was not on ADS-B; and appeared to make no attempt to avoid us. This event occurred on Day 0 at XA:30am.