  We establish various properties of the p-adic algebraic K-theory of smooth
algebras over perfectoid rings living over perfectoid valuation rings. In
particular, the p-adic K-theory of such rings is homotopy invariant, and
coincides with the p-adic K-theory of the p-adic generic fibre in high degrees.
In the case of smooth algebras over perfectoid valuation rings of mixed
characteristic the latter isomorphism holds in all degrees and generalises a
result of Nizio{\l}.
