  We investigate the finite-state $p$-solid-on-solid model, for $p=\infty$, on
Cayley trees of order $k\geq 2$ and establish a system of functional equations
where each solution corresponds to a (splitting) Gibbs measure of the model.
Our main result is that, for three states, $k=2,3$ and increasing coupling
strength, the number of translation-invariant Gibbs measures behaves as
$1\to3\to5\to6\to7$. This phase diagram is qualitatively similar to the one
observed for three-state $p$-SOS models with $p>0$ and, in the case of $k=2$,
we demonstrate that, on the level of the functional equations, the transition
$p\to\infty$ is continuous.
