  After some introductory discussion of the definition of Finsler spacetimes
and their symmetries, we consider a class of spherically symmetric and static
Finsler spacetimes which are small perturbations of the Schwarzschild
spacetime. The deviations from the Schwarzschild spacetime are encoded in three
perturbation functions $\phi_0(r)$, $\phi_1(r)$ and $\phi_2(r)$ which have the
following interpretations: $\phi_0$ perturbs the time function, $\phi_1$
perturbs the radial length measurement and $\phi_2$ introduces a spatial
anisotropy which is a genuine Finsler feature. We work out the equations of
motion for freely falling particles and for light rays, i.e. the timelike and
lightlike geodesics, in this class of spacetimes, and we discuss the bounds
placed on the perturbation functions by observations in the Solar system.
