  Colmez, Dospinescu and Niziol have shown that the only $p$-adic
representations of $\rm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$ appearing in the
$p$-adic \'etale cohomology of the coverings of Drinfeld's half-plane are the
$2$-dimensional cuspidal representations (i.e. potentially semi-stable, whose
associated Weil-Deligne representation is irreducible) with Hodge-Tate weights
$0$ and $1$ and their multiplicities are given by the $p$-adic Langlands
correspondence. We generalise this result to arbitrary weights, by considering
the $p$-adic \'etale cohomology with coefficients in the symmetric powers of
the universal local system on Drinfeld's tower. A novelty is the appearance of
potentially semistable $2$-dimensional non-cristabelian representations, with
expected multiplicity. The key point is that the local systems we consider turn
out to be particularly simple: they are "isotrivial opers" on a curve. We
develop a recipe to compute the pro\'etale cohomology of such a local system
using the Hyodo-Kato cohomology of the curve and the de Rham complex of the
flat filtered bundle associated to the local system.
