constant rank theorem for banach spaces

Yes, there is. The (Constant) Rank Theorem for Banach spaces is Theorem 2.5.15 of the book of R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications (3rd. edition, Springer-Verlag, 2001). There is a demand that the image of $DF[u_0]$ and the kernel of $DF[u_0]$ are closed direct summands for the $u_0\in B$ around which the theorem holds. The first requirement is automatic if $M$ is finite-dimensional.

There is also a version of the constant rank theorem in Glöckner's paper "Fundamentals of submersions and immersions between infinite-dimensional manifolds" (Theorem F of 1) which works specifically if the target is a finite-dimensional manifold (and the source an arbitrary manifold modeled on a locally convex space). The advantage of having a finite-dimensional target is that one can circumvent most of the tedious assumptions one needs for the case of an infinite-dimensional target (i.e. the ones from the version of Abraham, Marsden and Ratiu).

In 1 you can also find some references to constant rank theorems between Banach spaces. They are given after Theorem F.