  Let $F$ be a local field over $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$, and let
$D$ be a central simple division algebra over $F$ of degree $d$. In the
$p$-adic case, we assume $p>de+1$ where $e$ is the ramification degree over
$\mathbf{Q}_p$; otherwise, we need only assume $p$ and $d$ are coprime. For the
subgroup $I_1=1+\varpi_D \mathcal{O}_D$ of $D^\times$ we determine the
structure of $\mathrm{H}^1(I_1, \pi)$ as a representation of $D^\times/I_1$ for
an arbitrary smooth irreducible representation $\pi$ of $D^\times$. We use this
to compute the group $\mathrm{Ext}^1_{D^\times}(\pi,\pi')$ for arbitrary smooth
irreducible representations $\pi$ and $\pi'$ of $D^\times$. In the $p$-adic
case, via Poincar\'{e} duality we can compute the top cohomology groups and
compute the highest degree extensions.
