  We prove that the continuous group cohomology groups of a locally profinite
group $ G $ with coefficients in a smooth $ k $-representation $ \pi $ of $ G $
are isomorphic to the $ \mathrm{Ext}$-groups $ \mathrm{Ext}^i_G(\mathbb{1},\pi)
$ computed in the category of smooth $ k $-representations of $ G $. We apply
this to show that if $ \pi $ is a supersingular $ \overline{\mathbb{F}}_p
$-representation of $ \mathrm{GL}_2(\mathbb{Q}_p) $, then the continuous group
cohomology of $ \mathrm{SL}_2(\mathbb{Q}_p)$ with values in $ \pi $ vanishes.
Furthermore, we prove that the continuous group cohomology groups of a $ p
$-adic reductive group $ G $, with coefficients in an admissible unitary $
\mathbb{Q}_p $-Banach space representation $ \Pi $, are finite dimensional. We
show that the continuous group cohomology of $ \mathrm{SL}_2(\mathbb{Q}_p) $
with values in non-ordinary irreducible $ \mathbb{Q}_p $-Banach space
representations of $ \mathrm{GL}_2(\mathbb{Q}_p) $ vanishes.
