  Since their inception perfectoid spaces have catalyzed a revolution in p-adic
geometry. We redevelop the foundations of perfectoid spaces from the point of
view of Berkovich Spaces, where the underlying topological space of an affinoid
perfectoid space is a compact Hausdorff space -- closely resembling the
situation in complex geometry. The key technical ingredient in our construction
is arc_pi-descent for perfectoid Banach algebras. Along the way, we establish
various foundational results for arc_pi-sheaves, notably a form of the
Gerritzen-Grauert theorem.
