  Super-BMS$_4$ algebras -- also called BMS$_4$ superalgebras -- are graded
extensions of the BMS$_4$ algebra. They can be of two different types: they can
contain either a finite number or an infinite number of fermionic generators.
We show in this letter that, with suitable boundary conditions on the graviton
and gravitino fields at spatial infinity, supergravity on asymptotically flat
spaces possesses as superalgebra of asymptotic symmetries a (nonlinear)
super-BMS$_4$ algebra containing an infinite number of fermionic generators,
which we denote SBMS$_4$. These boundary conditions are not only invariant
under SBMS$_4$, but also lead to a fully consistent canonical description of
the supersymmetries, which have in particular well-defined Hamiltonian
generators that close according to the nonlinear SBMS$_4$ algebra. One finds in
particular that the graded brackets between the fermionic generators yield all
the BMS$_4$ supertranslations, of which they provide therefore "square roots".
