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