We prove that if a geodesically complete CAT(0) space X admits a
proper cocompact isometric action of a group, then the Izeki-Nayatani invariant
of X is less than 1. Let G be a finite connected graph, μ1​(G) be
the linear spectral gap of G, and λ1​(G,X) be the nonlinear spectral
gap of G with respect to such a CAT(0) space X. Then, the result
implies that the ratio λ1​(G,X)/μ1​(G) is bounded from below by a
positive constant which is independent of the graph G. It follows that any
isometric action of a random group of the graph model on such X has a global
fixed point. In particular, any isometric action of a random group of the graph
model on a Bruhat-Tits building associated to a semi-simple algebraic group has
a global fixed point