  Let $G$ be a split connected reductive group defined over a nonarchimedean
local field of residual characteristic $p$, and let $\mathcal{H}$ be the
pro-$p$-Iwahori--Hecke algebra associated to a fixed choice of pro-$p$-Iwahori
subgroup. We explore projective resolutions of simple right
$\mathcal{H}$-modules. In particular, subject to a mild condition on $p$, we
give a classification of simple right $\mathcal{H}$-modules of finite
projective dimension, and consequently show that "most" simple modules have
infinite projective dimension.
