  We prove that the Witt vector affine Grassmannian, which parametrizes
W(k)-lattices in W(k)[1/p]^n for a perfect field k of charactristic p, is
representable by an ind-(perfect scheme) over k. This improves on previous
results of Zhu by constructing a natural ample line bundle. Along the way, we
establish various foundational results on perfect schemes, notably h-descent
results for vector bundles.
