  We initiate the classification of nonrelativistic effective field theories
(EFTs) for Nambu-Goldstone (NG) bosons, possessing a set of redundant,
coordinate-dependent symmetries. Similarly to the relativistic case, such EFTs
are natural candidates for "exceptional" theories, whose scattering amplitudes
feature an enhanced soft limit, that is, scale with a higher power of momentum
at long wavelengths than expected based on the mere presence of Adler's zero.
The starting point of our framework is the assumption of invariance under
spacetime translations and spatial rotations. The setup is nevertheless general
enough to accommodate a variety of nontrivial kinematical algebras, including
the Poincare, Galilei (or Bargmann) and Carroll algebras. Our main result is an
explicit construction of the nonrelativistic versions of two infinite classes
of exceptional theories: the multi-Galileon and the multi-flavor
Dirac-Born-Infeld (DBI) theories. In both cases, we uncover novel Wess-Zumino
terms, not present in their relativistic counterparts, realizing nontrivially
the shift symmetries acting on the NG fields. We demonstrate how the symmetries
of the Galileon and DBI theories can be made compatible with a nonrelativistic,
quadratic dispersion relation of (some of) the NG modes.
