GA flight instructor reported a near miss on initial climb with an aircraft on final approach landing opposite direction at a non-towered airport. The instructor took control from the student and maneuvered away from the traffic.

Narrative

During a dual flight review at COD; Aircraft X experienced a near-midair collision with Aircraft Y on a short final to the opposite runway direction. No radio calls were heard from Aircraft Y on CTAF; and evasive action was required to avoid conflict.I was conducting a flight review at COD. We started the aircraft on the public ramp near the FBO. I served as a pilot monitoring (PM); responsible for radio communications and situational awareness.Before taxi; I announced on CTAF (122.8) our intention to taxi from the ramp to Runway 4 via Alpha. I monitored the frequency continuously; heard no other aircraft call; and scanned TCAS for traffic. There were no targets present.Approximately five minutes later; I announced our departure from Runway 4. As the pilot flying (PF) advanced the throttle; I began standard callouts and confirmed 'airspeed alive.' Suddenly; a TCAS target appeared directly ahead at 12 o'clock. I looked up and acquired visual contact with Aircraft Y on what seemed to be a short final to Runway 22; at approximately 150 feet AGL. We climbed through ~75 feet AGL on a direct collision course after takeoff.I immediately took control of the aircraft and turned left to avoid conflict. Aircraft Y did not appear to maneuver. I made repeated CTAF calls to them with no reply. After landing; Aircraft Y pilot called on CTAF; saying they could now hear me and were 'having trouble with the radio.'An airport operations supervisor; witnessed the event and confirmed over UNICOM that he saw the close call. He later spoke with the Aircraft Y crew; and filed a report.I've operated out of COD nearly daily for many years. This is not the first time I've observed aircraft arriving without making position reports or integrating safely with existing pattern traffic. I've seen student pilots forced to maneuver for uncoordinated aircraft arrivals.Contributing Factors: Lack of required CTAF communication from the Aircraft Y crew.Opposite-direction runway usage without coordination.Possible failure to monitor frequency.Lack of situational awareness on the part of Aircraft Y in non-towered operations.Human Factors: Communication Breakdown.Situational Awareness - Loss.Workload Management (PM/Instructor monitoring aircraft systems and radios). Corrective Actions / Recommendations:Emphasize proper use of CTAF per FAA.Reinforce company SOPs for those operating at non-towered fields.Improve CRM and radio cross-checks before the final approach into uncontrolled airports.Consider simulator-based training on pattern entry and coordination at non-towered airports.Intent of Report: To highlight recurring communication and coordination issues with aircraft operating at non-towered fields and to encourage improved adherence to CTAF procedures for public and pilot safety.