  We prove that the infinite family of asymptotic mapping class groups of
surfaces of defined by Funar--Kapoudjian and Aramayona--Funar are of type
$F_\infty$, thus answering questions of Funar-Kapoudjian-Sergiescu and
Aramayona-Vlamis.
  As it turns out, this result is a specific instance of a much more general
theorem which allows to deduce that asymptotic mapping class groups of Cantor
manifolds, also introduced in this paper, are of type $F_\infty$, provide the
underlying manifolds satisfy some general hypotheses.
  As important examples, we will obtain $F_\infty$ asymptotical mapping class
groups that contain, respectively, the mapping class group of every compact
surface with non-empty boundary, the automorphism group of every free group of
finite rank, or infinite families of arithmetic groups.
  In addition, for certain types of manifolds, the homology of our asymptotic
mapping class groups coincides with the stable homology of the relevant mapping
class groups, as studied by Harer and Hatcher--Wahl.
