  Let $G$ be a split connected reductive group over a finite extension $F$ of
$Q_p$. We determine the extensions between unitary continuous $p$-adic and
smooth mod $p$ principal series of $G(F)$ in the generic case. In order to do
so, we compute Emerton's delta-functor $\mathrm{H^\bullet Ord}_{B(F)}$ of
derived ordinary parts with respect to a Borel subgroup on certain induced
representations of $G(F)$ using a Bruhat filtration. These extensions come into
play in the $p$-adic and mod $p$ Langlands programs.
