  We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian
and the corresponding eigenfunctions upon perturbation of the domain and we
generalize a few results known for the standard $p$-Laplacian. In particular,
we prove a Frech\'{e}t differentiability result for the eigenvalues, we compute
the corresponding Hadamard formulas and we prove a continuity result for the
eigenfunctions. Finally, we briefly discuss a well-known overdetermined problem
and we show how to deduce the Rellich-Pohozaev identity for the Finsler
$p$-Laplacian from the Hadamard formula.
