  We give a new construction of $p$-adic overconvergent Hilbert modular forms
by using Scholze's perfectoid Shimura varieties at infinite level and the
Hodge--Tate period map. The definition is analytic, closely resembling that of
complex Hilbert modular forms as holomorphic functions satisfying a
transformation property under congruence subgroups. As a special case, we first
revisit the case of elliptic modular forms, extending recent work of Chojecki,
Hansen and Johansson. We then construct sheaves of geometric Hilbert modular
forms, as well as subsheaves of integral modular forms, and vary our
definitions in $p$-adic families. We show that the resulting spaces are
isomorphic as Hecke modules to earlier constructions of Andreatta, Iovita and
Pilloni. Finally, we give a new direct construction of sheaves of arithmetic
Hilbert modular forms, and compare this to the construction via descent from
the geometric case.
