  Let $G$ be a $p$-adic reductive group and $\mathfrak{g}$ its Lie algebra. We
construct a functor from the extension closure of the Bernstein-Gelfand-Gelfand
category $\mathcal{O}$ associated to $\mathfrak{g}$ into the category of
locally analytic representations of $G$, thereby expanding on an earlier
construction of Orlik-Strauch. A key role in this new construction is played by
$p$-adic logarithms on tori. This functor is shown to be exact with image in
the subcategory of admissible representations in the sense of Schneider and
Teitelbaum. En route, we establish some basic results in the theory of modules
over distribution algebras and related subalgebras, such as a tensor-hom
adjunction formula. We also relate our constructions to certain representations
constructed by Breuil and Schraen in the context of the $p$-adic Langlands
program.
