  We prove that the categories of smooth and analytic unitary representations
of Banach--Lie supergroups are well-behaved under restriction functors, in the
sense that the restriction of a representation to an integral subsupergroup is
well-defined. We also prove that the category of analytic representations is
isomorphic to a subcategory of the category of smooth representations. These
facts are needed as a crucial first step to a rigorous treatment of the
analytic theory of unitary representations of Banach--Lie supergroups. They
extend the known results for finite dimensional Lie supergroups. In the
infinite dimensional case the proofs require several new ideas. As an
application, we give an analytic realization of the oscillator representation
of the restricted orthosymplectic Banach--Lie supergroup.
