Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-integrable functions from X to Y.
Let $\operatorname{Lip}_c(X,Y)$ denote the set of Lipschitz functions from $X$ to $Y$ with closed and bounded support.
If $1 \le p< \infty$ then is $\operatorname{Lip}_c(X,Y)$ dense in $L^p(X,Y;\mu)$?
If not, then what if we additionally assume that X and Y are Hilbert spaces?