  The pro-$p$ Iwahori-Hecke $\operatorname{Ext}$-algebra $E^\ast$ is a graded
algebra that has been introduced and studied by Ollivier-Schneider, with the
long-term goal of investigating the category of smooth mod-$p$ representations
of $p$-adic reductive groups and its derived category. Its $0$th graded piece
is the pro-$p$ Iwahori-Hecke algebra studied by Vign\'eras and others.
  In the present article, we first show that the $\operatorname{Ext}$-algebra
$E^\ast$ associated with the group $\operatorname{SL}_2(\mathfrak{F})$,
$\operatorname{PGL}_2(\mathfrak{F})$ or $\operatorname{GL}_2(\mathfrak{F})$,
where $\mathfrak{F}$ is an unramified extension of $\mathbb{Q}_p$ with $p \neq
2,3$, is finitely generated as a (non-commutative) algebra.
  We then specialize to the case of the group
$\operatorname{SL}_2(\mathbb{Q}_p)$, with $p \neq 2,3$, and we show that in
this case the natural multiplication map from the tensor algebra $T^\ast_{E^0}
E^1$ to $E^\ast$ is surjective and that its kernel is finitely generated as a
two-sided ideal. Using this fact as main input, we then show that $E^\ast$ is
finitely presented as an algebra. We actually compute an explicit presentation.
