Flight Instructor reported a NMAC in the traffic pattern at P08 non-towered airport. The conflicting aircraft took evasive action to avoid a collision.

Narrative

At approximately XA:50; Aircraft X; was conducting landing practice at Coolidge Airport (P08). At XA:56; Aircraft X opted to extend their downwind leg to allow for traffic on the ground to depart prior to their arrival. Position reports were made accordingly. After extending; at XA:56; Aircraft X turned an extended right base for the runway in use; Runway 23 - and continued making position reports. Up until this point; Aircraft Y had not been making position reports; other than that they were inbound on the GPS approach for Runway 23. Aircraft Y did not state their type of aircraft; or that they were requesting priority landing. Aircraft X assumed there would be adequate spacing when they made their base turn. Between XA:58 - XB:00; Aircraft Y continued to fly their practice instrument approach; not yielding to traffic established in the pattern; in direct violation of the advice set forth in AC 90-66. At XB:00; Aircraft Y opted to not stop on the runway; (which would have given Aircraft X space); but instead continued at a high rate of speed until XB:01 where it became necessary for them to off-set their course to the left to avoid hitting Aircraft X. At XB:02; Aircraft X departed the pattern out of an abundance of caution.