  We consider a non-relativistic limit of the bosonic sector of
eleven-dimensional supergravity, leading to a theory based on a covariant
`membrane Newton-Cartan' (MNC) geometry. The local tangent space is split into
three `longitudinal' and eight `transverse' directions, related only by
Galilean rather than Lorentzian symmetries. This generalises the
ten-dimensional stringy Newton-Cartan (SNC) theory. In order to obtain a finite
limit, the field strength of the eleven-dimensional four-form is required to
obey a transverse self-duality constraint, ultimately due to the presence of
the Chern-Simons term in eleven dimensions. The finite action then gives a set
of equations that is invariant under longitudinal and transverse rotations,
Galilean boosts and local dilatations. We supplement these equations with an
extra Poisson equation, coming from the subleading action. Reduction along a
longitudinal direction gives the known SNC theory with the addition of RR gauge
fields, while reducing along a transverse direction yields a new
non-relativistic theory associated to D2 branes. We further show that the MNC
theory can be embedded in the U-duality symmetric formulation of exceptional
field theory, demonstrating that it shares the same exceptional Lie algebraic
symmetries as the relativistic supergravity, and providing an alternative
derivation of the extra Poisson equation.
