  This belongs to a series of papers motivated by Ballmann's Higher Rank
Rigidity Conjecture. We prove the following. Let $X$ be a CAT(0) space with a
geometric group action. Suppose that every geodesic in $X$ lies in an $n$-flat,
$n\geq 2$. If $X$ contains a periodic $n$-flat which does not bound a flat
$(n+1)$-half-space, then $X$ is a Riemannian symmetric space, a Euclidean
building or non-trivially splits as a metric product. This generalizes the
Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group
actions.
