Let $X$ be a separable Banach space embedded canonically in $X^{\*\*}$. Is there a retraction from the unit ball $B\_{X^{\*\*}}$ of $X^{\*\*}$ onto the unit ball $B\_X$ of $X$?


When we insist on uniformly continuous retraction, the answer is no, I think, but what if we simply ask for continuous retractions?



Are unit balls in Banach spaces retracts of bidual balls?
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](https://projecteuclid.org/euclid.pjm/1103052106). 


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^{\*\*}}}\, $.


