  We define a (perfectoid) mixed characteristic version of $F$-signature and
Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of
Bhatt-Scholze and Faltings' normalized length (also developed in the work of
Gabber-Ramero). We show that these definitions coincide with the classical
theory in equal characteristic $p > 0$. We prove that a ring is regular if and
only if either its perfectoid signature or perfectoid Hilbert-Kunz multiplicity
is 1 and we show that perfectoid Hilbert-Kunz multiplicity characterizes BCM
closure and extended plus closure of $\mathfrak{m}$-primary ideals. We
demonstrate that perfectoid signature detects BCM-regularity and transforms
similarly to $F$-signature or normalized volume under quasi-\'etale maps. As a
consequence, we prove that BCM-regular rings have finite local \'etale
fundamental group and also finite torsion part of their divisor class groups.
Finally, we also define a mixed characteristic version of relative rational
signature, and show it characterizes BCM-rational singularities.
