A pilot reported a NMAC on base leg to a non-towered airport with an aircraft also on base leg.

Narrative

I entered downwind at 45 degree for 26 runway at 3CK with radio CTAF announcement before entering and when established on the 26 runway left downwind. In front of me there was traffic (Aircraft Z) and I slowed down to 80mph to let that traffic take base 26 first. I did extend the downwind to give more space for landing traffic. As the traffic was established on final 26 I started turning base 26 and as I looked at airport I was surprised that Aircraft Y that was right behind me and also turning base 26. As soon I spotted that I stopped turning base 26; I radio my surprise of the traffic on base 26 and flew straight making announcement on CTAF about my intention of giving space via 360 right turn on base to give space that unexpected traffic Aircraft Y. Other traffic airplanes that were on downwind also extended their turn to base. To prevent this high traffic situation; all pilots should announce and listen other pilots; to visualize their position and better check for traffic on same pattern leg.