We consider Newton's method for finding zeros of mappings from a manifold 𝒳 into a vector bundle ℰ. In this setting a connection on ℰ is required to render the Newton equation well defined, and a retraction on 𝒳 is needed to compute a Newton update. We discuss local convergence in terms of suitable differentiability concepts, using a Banach space variant of a Riemannian distance. We also carry over an affine covariant damping strategy to our setting. Finally, we will illustrate our results by ...