This seems plausible, given the properties of the unit ball of $c_0$.
I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be nice to deduce from this that $X$ contains no copy of $c_0$. I have been searching, but could find no proof either way, and I cannot see how to prove it myself---well, not yet... But in the mean time, perhaps this is already known.