PA28 flight instructor reported an NMAC event during initial departure with a none reporting opposite direction landing aircraft. Flight instructor executed an evasive maneuver to avoid a collision.

Narrative

At XA:52 on Date our aircraft a Piper Warrior was departing from ZZZ VFR to ZZZ1. We were on the CTAF frequency making all of our taxi; and departing calls. We taxied to Runway X after receiving the weather; the winds were light but favoring Runway X. We taxied without incident; while holding short someone did ask on CTAF if the tower was open; we replied that they were not open until XB:00. We conducted a run up and briefed a short field takeoff. We made our call that we were taking the active Runway X; with no one else answering back; we began our short field takeoff; while on climb out our ADS-B indicated there was an aircraft coming in on the opposite Runway XX; NOT making any calls. I then announced on CTAF that we were in the upwind for Runway X. Still no other call from the incoming traffic. That's when the aircraft appeared to be approaching at a high speed; our aircraft was approximately 600-800 ft. MSL when we made our abrupt turn to the left announcing on CTAF that we were making our turn to the left hoping the other traffic would call in. As we made that turn I looked to my right and saw the aircraft pass under us about 100 feet or less. The aircraft finally made a call saying 'yeah we were on the wrong frequency'. The other aircraft's tail number is Aircraft Y and appeared to be a King Air. No other call on the CTAF was made from the King Air; exiting the runway or taxiing to the FBO. We made our call that we were departing the area to the NW. Since then I called tower to confirm if the radio transmissions were recorded; they stated they should be that we could go through the channels to pull the tapes.