Suppose that $Y$ is an infinite dimensional Banach space, with an embedded finite dimensional compact submanifold $X$. It is well known (cfr Lang's Differential and Riemannian Manifolds, Thm. 5.1) that $X$ has a tubular neighborhood in $Y$, consisting in a vector bundle $\pi: NX \to X$ and a homeomorphism $f: Z \to U$, where $Z$ is a neighborhood of the zero section $\zeta X$ in $NX$ and $X \subset U \subset Y$ is an open set, satifying $f = \zeta^{-1} \circ \iota$, where $\iota: X \to Y$ is the canonical inclusion.
Under these hypothesis, is it true that a closest point projection from $U$ to $Z$ is well-defined?