  We study a non-relativistic limit of 11-dimensional supergravity. This limit
leads to a theory with an underlying membrane Newton-Cartan geometry.
Consistency of the non-relativistic limit requires the imposition of
constraints, requiring that certain bosonic and fermionic torsions and
curvatures vanish. We investigate the implications of two versions of these
constraints. In one version, we keep only 16 supersymmetry transformations,
leading to a simple (purely bosonic) constraint structure but an unusual
realisation of the supersymmetry algebra which does not close into
diffeomorphisms. In the other, we keep all 32 supersymmetry transformations.
This requires a complicated sequence of bosonic and fermionic constraints,
eventually involving three derivatives of bosonic fields. We argue, with a
linearised calculation, that this sequence of constraints terminates.
Furthermore, we show that there exists a family of supersymmetric solutions
satisfying these constraints, containing the non-relativistic limit of the M2
supergravity solution recently obtained by Lambert and Smith as a background
relevant for non-relativistic holography.
