The question is the title. The set $Ba(C(K))$ is the unit ball of $C(K)$. This has to be known, but I can't find the answer explicitly in the literature. There is some literature about polyhedral Banach spaces however I'm not sure if being polyhedral is sufficient to show the set of extreme points of the ball is countable. Note that these $C(K)$ spaces are isomorphic to $C(\omega^{\omega^\alpha}+1)$ for some $\alpha$.

The motivation for the question is that I spent some time proving each Banach space in certain class has countably many extreme points. I then discovered that each of these space isometrically embed into $C(K)$ for some countable $K$.