  We study the basic geometry of a class of analytic adic spaces that arise in
the study of the extended (or adic) eigenvarieties constructed by
Andreatta--Iovita--Pilloni, Gulotta and the authors. We apply this to prove a
general interpolation theorem for Langlands functoriality, which works for
extended eigenvarieties and improves upon existing results in characteristic 0.
As an application, we show that the characteristic p locus of the extended
eigenvariety for GL(2)/F, where F is a cyclic extension of the rational numbers
Q, contains non-ordinary components of dimension at least [F:Q].
