Arc-descent for the perfect loop functor and $p$-adic Deligne--Lusztig spaces

We prove that the perfect loop functor $LX$ of a quasi-projective scheme $X$ over a local non-archimedean field $k$ satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group $G$, the map $LG\to L(G\mathrm{/}B)$ is a $v$-surjection. This gives a mixed characteristic version (for $v$-topology) of an equal characteristic result (in 'etale topology) of Bouthier–\v{C}esnavi\v{c}ius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne–Lusztig spaces $X_w(b)$ attached to unramified $p$-adic reductive groups. We show that in various cases these sheaves are ind-representable, thus partially solving a question of Boyarchenko. Finally, we show that the natural covering spaces ${\dot{X}}_{\dot{w}}(b)$ are pro-'etale torsors over clopen subsets of $X_w(b)$, and analyze some examples.