  Rapoport-Zink spaces are deformation spaces for $p$-divisible groups with
additional structure. At infinite level, they become preperfectoid spaces. Let
$\mathscr{M}_{\infty}$ be an infinite-level Rapoport-Zink space of EL type, and
let $\mathscr{M}_{\infty}^\circ$ be one geometrically connected component of
it. We show that $\mathscr{M}_{\infty}^{\circ}$ contains a dense open subset
which is cohomologically smooth in the sense of Scholze. This is the locus of
$p$-divisible groups which do not have any extra endomorphisms. As a corollary,
we find that the cohomologically smooth locus in the infinite-level modular
curve $X(p^\infty)^{\circ}$ is exactly the locus of elliptic curves $E$ with
supersingular reduction, such that the formal group of $E$ has no extra
endomorphisms.
