Sobolev mappings: Lipschitz density is not an isometric invariant of the target

If $M$ is a compact smooth manifold and $X$ is a compact metric space, the Sobolev space $W^{1,p}(M,X)$ is defined through an isometric embedding of $X$ into a Banach space. We prove that the answer to the question whether Lipschitz mappings ${\rm Lip}\,(M,X)$ are dense in $W^{1,p}(M,X)$ may depend on the isometric embedding of the target.