  We develop the $p$-adic representation theory of $p$-adic Lie groups on solid
vector spaces over a complete non-archimedean extension of $\mathbf{Q}_p$. More
precisely, we define and study categories of solid, solid locally analytic and
solid smooth representations. We show that the category of solid locally
analytic representations of a compact $p$-adic Lie group is equivalent to that
of quasi-coherent modules over its algebra of locally analytic distributions,
generalizing a classical result of Schneider and Teitelbaum. For arbitrary $G$,
we prove an equivalence between solid locally analytic representations and
quasi-coherent sheaves over certain locally analytic classifying stack of $G$.
We also extend our previous cohomological comparison results from the case of a
compact group defined over $\mathbf{Q}_p$ to the case of an arbitrary group,
generalizing results of Lazard and Casselman-Wigner. Finally, we study an
application to the locally analytic $p$-adic Langlands correspondence for
$\mathrm{GL}_1$.

Representation Theory;Number Theory