Growth rate of Lipschitz constants for retractions between finite subset spaces

For any metric space $X$, finite subset spaces of $X$ provide a sequence isometric embeddings $X=X(1)\subset X(2)\subset\cdots$. The existence Lipschitz retractions $r_n\colon X(n)\to X(n-1)$ depends on the geometry in subtle way. Such are known to exist when is an Hadamard or finite-dimensional normed space. But even these cases it was unknown whether $\{r_n\}$ can be uniformly Lipschitz. We give negative answer by proving that $\operatorname{Lip}(r_n)$ must grow with $n$