  We prove that if a geodesically complete $\mathrm{CAT}(0)$ 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, $\mu_1 (G)$ be
the linear spectral gap of $G$, and $\lambda_1 (G,X)$ be the nonlinear spectral
gap of $G$ with respect to such a $\mathrm{CAT}(0)$ space $X$. Then, the result
implies that the ratio $\lambda_1 (G,X) / \mu_1 (G)$ 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.
