$\DeclareMathOperator\Tub{Tub}$I have not found any reference among the well-known books about the existence of a geometric tubular neighborhood in the Finsler spaces. I am wondering if there exists such a neighborhood for any closed submanifold of a Fisnler Manifold $M$ (maybe with some extra hypothesis on $M$ ,like compactness ....).
FYI: by a geometric tubular neighborhood, of a pre-compact submanifold of a manifold $M$, I mean: $\Tub(P)_r:=\{\gamma(1)|\gamma:[0,1]\longrightarrow M $ is a minimizing geodesic with $\gamma'(0)\in\mathfrak{C}_{\gamma(0)}(P)\cap B(r)\}$. where $B(r)$ is the ball of radius $r$ and by $\mathfrak{C}_x$ we mean the subset of vectors that each one is orthogonal to $T_xP$ at the direction of itself.