  We analyze the geometry of some $p$-adic Deligne--Lusztig spaces $X_w(b)$
introduced in [Iva21] attached to an unramified reductive group ${\bf G}$ over
a non-archimedean local field. We prove that when ${\bf G}$ is classical, $b$
basic and $w$ Coxeter, $X_w(b)$ decomposes as a disjoint union of translates of
a certain integral $p$-adic Deligne--Lusztig space. Along the way we extend
some observations of DeBacker and Reeder on rational conjugacy classes of
unramified tori to the case of extended pure inner forms, and prove a loop
version of Frobenius-twisted Steinberg's cross section.
