NOTE: I am very much not an expert on affine Kac-Moody groups or ind-varieties, so the following may be vague.

The first question is: Has anyone developed a theory of Frobenius splitting for ind-varieties in positive characteristic? (This might be very easy for all I know). Following from that, my main question is: If there is a reasonable theory of Frobenius splitting of ind-varieties, has anyone thought about whether affine flag varieties or affine Grassmannians corresponding to semisimple algebraic groups in positive characteristic are Frobenius split?