  Let $G$ be a group with a non-elementary action on a proper CAT(0) space $X$,
and let $\mu$ be a measure on $G$ such that the random walk $(Z_n)_n$ generated
by $\mu$ has finite second moment on $X$. Let $o$ be a basepoint in $X$, and
assume that there exists a rank one isometry in $G$. We prove that in this
context, $(Z_n o )_n$ satisfies a Central Limit Theorem, namely that the random
variables $\frac{1}{\sqrt{n}}(d(Z_n o, o) - n \lambda) $ converge in law to a
non-degenerate Gaussian distribution $N_\mu$, for $\lambda$ the (positive)
drift of the random walk. The strategy relies on the use of hyperbolic models
introduced by H. Petyt, A. Zalloum and D. Spriano, which are analogues of curve
graphs and cubical hyperplanes for the class of CAT(0) spaces. As a side
result, we prove that the probability that the nth-step $Z_n$ acts on $X$ as a
contracting isometry goes to 1 as $n$ goes to infinity.
