  Let $f \colon X \dashrightarrow X$ be a dominant rational self-map of a
smooth projective variety defined over $\overline{\mathbb Q}$. For each point
$P\in X(\overline{\mathbb Q})$ whose forward $f$-orbit is well-defined,
Silverman introduced the arithmetic degree $\alpha_f(P)$, which measures the
growth rate of the heights of the points $f^n(P)$. Kawaguchi and Silverman
conjectured that $\alpha_f(P)$ is well-defined and that, as $P$ varies, the set
of values obtained by $\alpha_f(P)$ is finite. Based on constructions of
Bedford--Kim and McMullen, we give a counterexample to this conjecture when
$X=\mathbb P^4$.
