# Rademacher‘s Theorem and Baire category

Is there a variant of Rademacher‘s Theorem where the smallness of the points of non-differentiability is measured in terms of Baire category instead of measure?

More precisely, let X be a separable Banach space and Y be a space with the RNP. Moreover, let $U\subset X$ be an open set and $f\colon U\to Y$ be a Lipschitz mapping. Is the set of points where $f$ is not differentiable a meager set?

As a partial question: Is something known about the above in finite dimensions?