  Let $G$ be a split connected reductive group over a non-archimedean local
field. In the $p$-adic setting, Orlik-Strauch constructed functors from the BGG
category $\mathcal{O}$ associated to the Lie algebra of $G$ to the category of
locally analytic representation of $G$. We generalize these functors to such
groups over non-archimedean local fields of arbitrary characteristic. To this
end, we introduce the hyperalgebra of a non-archimedean Lie group $G$, which
generalizes its Lie algebra, and consider topological modules over the algebra
of locally analytic distributions on $G$ and subalgebras related to this
hyperalgebra.
