  Let $\mathfrak{F}$ be a nonarchimedean local field of residual characteristic
$p$, and let $G$ denote the group of $\mathfrak{F}$-points of a connected
reductive group over $\mathfrak{F}$. For an open compact subgroup $\mathcal{U}$
of $G$ and a unital commutative ring $k$, we let $\mathbf{X}_{\mathcal{U}}$
denote the space of compactly supported $k$-valued functions on
$G/\mathcal{U}$. Building on work of Ollivier--Schneider, we investigate the
graded $\textrm{Ext}$-algebra $E_{\mathcal{U}}^* :=
\textrm{Ext}_G^*(\mathbf{X}_{\mathcal{U}},\mathbf{X}_{\mathcal{U}})^{\textrm{op}}$.
In particular, we describe the Yoneda product, an involutive anti-automorphism,
and (when $k$ is a field of characteristic $p$ and $\mathcal{U}$ has no
$p$-torsion) a duality operation. We allow for the reductive group to be
non-split, and for the open compact subgroup $\mathcal{U}$ to be non-pro-$p$.
  Specializing further to the case $G = \textrm{SL}_2(\mathbb{Q}_p)$ with $p
\geq 5$ and a coefficient field of characteristic $p$, we obtain more precise
results when $\mathcal{U}$ is equal to an Iwahori subgroup $J$ or a
hyperspecial maximal compact subgroup $K$. In particular, we compute the
structure of $E_J^*$ as an $E_J^0$-bimodule, obtain an explicit description of
the center $\mathcal{Z}(E_J^*)$ of $E_J^*$, and construct a surjective morphism
of algebras $\mathcal{Z}(E_J^*) \longrightarrow E_K^*$ (analogous to the
compatibility between Bernstein and Satake isomorphisms in characteristic 0).
From this we deduce the (somewhat surprising) fact that $E_K^*$ is not
graded-commutative, contrary to what happens for almost all $\ell$-modular
characteristics.
