  A relevant property of equifocal submanifolds is that their parallel sets are
still immersed submanifolds, which makes them a natural generalization of the
so-called isoparametric submanifolds. In this paper, we prove that the regular
fibers of an analytic map $\pi:M^{m+k}\to B^{k}$ are equifocal whenever
$M^{m+k}$ is endowed with a complete Finsler metric and there is a restriction
of $\pi$ which is a Finsler submersion for a certain Finsler metric on the
image. In addition, we prove that when the fibers provide a singular foliation
on $M^{m+k}$, then this foliation is Finsler.
