  The classical Brody's theorem asserts the equivalence between two notions of
hyperbolicity for compact complex spaces, one named after Kobayashi and one
expressed in terms of lack of non constant holomorphic entire functions
(compactness is only used to prove the harder implication). We extend this
theorem to Deligne-Mumford analytic stacks, by first providing definitions of
what we think of Kobayashi and Brody hyperbolicity for such objects and then
proving the equivalence of these concepts under an assumption of compactness.
