  In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups
$\Bbb{I}$ for large enough $p$. This is done by endowing $I$ with a natural
$p$-valuation, and thereby generalizing results of Lazard for $\text{GL}_n$. We
work with a general connected reductive split group over some $p$-adic field
(with simply connected derived group) and study the $\Bbb{I}$-analytic vectors
in principal series representations. Our main result is an irreducibility
criterion which generalizes results of Clozel and Ray in the
$\text{GL}_n$-case.
