  We study Finsler spacetimes and Killing vector fields taking care of the fact
that the generalized metric tensor associated to the Lorentz-Finsler function
$L$ is in general well defined only on a subset of the slit tangent bundle. We
then introduce a new class of Finsler spacetimes endowed with a timelike
Killing vector field that we call stationary splitting Finsler spacetimes. We
characterize when a Finsler spacetime with a timelike Killing vector field is
locally a stationary splitting. Finally, we show that the causal structure of a
stationary splitting is the same of one of two Finslerian static spacetimes
naturally associated to the stationary splitting.
