A geodesic metric space is a manifold on its own right. What are conditions for a Finsler space to be a manifold?

Question

A geodesic metric space can locally be approximated by a vector space. This approximation provides it with a natural manifold structure. It means that geodesic metric space is more fundamental concept and to be a manifold is just a quality of it.
Talking about a Finsler space for example, which is a metric space by construction, we would not better call it a manifold upfront, but instead check the conditions for it to be a manifold.
Can we say what those conditions are for a Finsler space to be a manifold? Or is it so that basically any Finsler space is always a manifold and that is the reason why they are so often called simply Finsler manifolds?

Answer 1

In the Riemannian setting, this question is answered in detail in my answer here. The problem for Finsler manifolds is wide-open (to the best of my knowledge).