A Separable Banach Space with a Schauder Basis Which Is Not a Lipschitz Retract of Its Bidual

We construct a separable Banach space $X$ with a Schauder basis such that $B_X$ is not a uniformly continuous retract of $B_{X^{\ast \ast }}$. Consequently, $X$ is not a uniformly continuous retract of $X^{\ast \ast }$ and hence is not a Lipschitz retract of $X^{\ast \ast }$.