  Finsler geometry is a natural arena to investigate the physics of spacetimes
with local Lorentz violating. The directional dependence of the Finsler metric
provides a way to encode the Lorentz violating effects into the geometric
structure of spacetime. Here, a classical field theory is proposed in a special
Finsler geometry, the so-called Randers-Finsler spacetime, where the Lorentz
violation is produced by a background vector field. By promoting the
Randers-Finsler metric to a differential operator, a Finsler-invariant action
for the scalar, gauge and fermions are proposed. The theory contains nonlocal
terms, as in the Very Special Relativity based theories. By expanding the
Lagrangian, minimal and nonminimal Standard Model Extension terms arises,
revealing a perturbative Lorentz violation. For a CPT-even term, the
Carrol-Field-Jackiw and derivative extensions are obtained.
