Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces

We show that if $Y$ is a separable subspace of a Banach space $X$such that both $X$ and the quotient $X/Y$ have $C^p$-smooth Lipschitz bump functions, and $U$ is a bounded open subset of $X$, then, for every uniformly continuous function $f:Y\cap U\to\mathbb{R}$ and every $\varepsilon>0$, there exists a $C^p$-smooth Lipschitz function $F:X\to\mathbb{R}$ such that $\vert F(y)-f(y)\vert\leq\varepsilon$ for every $y\in Y\cap U$.