We explain that extremal weight modules of quantum loop algebras give rise to the projective coordinate ring of the formal model of the semi-infinite flag manifolds over the ring of integers with two inverted. Then, we exhibit how this gives rise to the Frobenius splitting of such an (ind-)scheme. This particularly implies that the Schubert varieties of the quasi-map spaces from a projective line to a (partial) flag manifold admits a Frobenius splitting compatible with the boundaries, and consequently such varieties are normal and has rational singularity in characteristic zero. This extends the case of the genuine quasi-map spaces by Braverman-Finkelberg and the asymptotic case by myself.
If time allows, we explain how to use such results to understand the structure of equivariant small quantum $K$-theory of a (partial) flag manifold.