  We extend the work of Pappas-Rapoport-Zhu on twisted affine Grassmannians to
wildly ramified, quasi-split, and residually split groups, assuming the maximal
torus is induced. This relies on the construction, inspired by Tits, of certain
smooth, affine, and connected $\mathbb{Z}[t]$-groups of parahoric type, which
should be regarded as $\mathbb{Z}$-families of parahoric group schemes, and
naturally extends a similar construction in the above articles after inverting
$e$. The resulting $\mathbb{F}_p(t)$-groups are pseudo-reductive possibly
non-standard in the sense of Conrad--Gabber--Prasad, and their
$\mathbb{F}_p[[t]]$-models are parahoric in our generalized sense. We study
their affine Grassmannians, establishing normality of Schubert varieties and
Zhu's coherence theorem.
