  We consider simple scalar theories with quadratic terms that are nonlocal and
Lorentz violating. Unlike similar Lorentz-invariant nonlocal theories that we
have considered previously, the theories studied here are both ghost-free and
unitary as formulated in Minkowski space. We explore the possibility that the
scale of nonlocality could be low in a dark sector, where the stringent
constraints on the violation of Lorentz invariance may be accommodated via the
weak coupling to the standard model. We point out that long-range forces may
originate from such a sector and be distinguishable from more conventional
beyond-the-standard model possibilities. We present a model in which a
nonlocal, Lorentz-violating dark sector communicates with the standard model
via a sector of heavy vector-like fermions, a concrete framework in which
phenomenological constraints and signals can be investigated.
