Indeeed, if $B\subset C$ are closed convex subsets of a Banach space, like in this case, the identity map $B\to B$ extends to a continuous retraction $C\to B$, by the Dugundji extension theorem.

Indeed, if $B\subset C$ are closed convex subsets of a Banach space, like in this case, the identity map $B\to B$ extends to a continuous retraction $C\to B$, by the Dugundji extension theorem.

Since a Banach space has Lipschitz partition of unity, the resulting retraction can be made locally Lipschitz.

Also note that if in your setting the Banach space is itself is a dual, $X=Y^*$, then
there is a norm-one linear projector $P$ of $X^{**}$ onto $X$ given by the composition of $\iota_{Y}^*: Y^{***}\to Y^*$ with $ \iota_{Y^*}:Y^* \to Y^{***}$. So in this case a retraction of $B_{X^{**}}$ onto $B_{X}$ is just $P_{|B_{X^{**}}}\, $.