Let $X$ be a separable pointed metric space and let $AE(X)$ denote the corresponding Lipschitz-Free (or Arens-Eells space) over $X$. The point-evaluation map $\delta:X\mapsto AE(X)$ is injective hence it has a left-inverse. When $X$ is Banach, the left-inverse may be taken to be continuous and linear.

In general, if $X$ is connected but not necessarily a topological vector space, does a continuous left-inverse of $\delta$ exist?