On the $L$-embeddability of Lipschitz-free spaces in their biduals

A Banach space $X$ is $L$-embedded in its bidual if there is a projection $P\text{\hspace{0.17em}⁣}:X^{\ast \ast }\to X$ such that $\mathrm{\parallel }x\mathrm{\parallel }=\mathrm{\parallel }Px\mathrm{\parallel }+\mathrm{\parallel }x-Px\mathrm{\parallel }$ for all $x\in X^{\ast \ast }$. We show that, as long as $X$ has dimension at least $2$, its Lipschitz-free space, denoted by $\mathcal{F}(X)$, is not $L$-embedded in its bidual. Our methods have applications to the problem of when the $L$-embeddability of $\mathcal{F}(M)$ passes to $\mathcal{F}(A)$ for a metric space $M$ and $A\subseteq M$. This is the case when $A$ is compact or when $A$ is geodesically closed.