Let $X$ be a smooth manifold and $Y$ be a closed submanifold. Then there exists a neighbourhood $U$ of $Y$ in $X$ such that $Y$ is a deformation retract of $U$ right?
I can only find (stronger forms of) this in literature under the assumption that $Y$ is compact, which I don't think is necessary for the above statement. So what would be a reference?
Since you just want a reference request: see Theorem III.2.2, Corollary III.2.3, and the remark after Definition III.2.4 of Kosinski's Differential Manifolds.