We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian manifold.
Title: Any sub-Riemannian metric has points of smoothness
Authors: A.A. Agrachev
Journal title: Russian Math. Dokl.
MathNet (ENG): http://mi.mathnet.ru/eng/dan153