Transfinite contractive propagation and the ball fixed point property in \texorpdfstring{$C(K)$}{C(K)}

We prove that, for a compact Hausdorff space $K$, the real Banach space $C(K)$ has the ball fixed point property if and only if $K$ is extremally disconnected. The new implication is obtained by constructing a fixed-point-free nonexpansive self-map of the whole closed unit ball whenever $K$ is a com