  If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nested
sequence under natural isometric embeddings $X = X(1)\subset X(2) \subset
\cdots$. It was previously established, by Kovalev when $X$ is a Hilbert space
and, by Ba\v{c}\'{a}k and Kovalev when $X$ is a CAT(0) space, that this
sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ for all $n\geq
2$. We prove that when $X$ is a normed space, the above sequence admits
Lipschitz retractions $X(n)\rightarrow X$, $X(n)\rightarrow X(2)$, as well as
concrete retractions $X(n)\rightarrow X(n-1)$ that are Lipschitz if $n=2,3$ and
H\"older-continuous on bounded sets if $n>3$. We also prove that if $X$ is a
geodesic metric space, then each $X(n)$ is a $2$-quasiconvex metric space.
These results are relevant to certain questions in the aforementioned previous
work which asked whether Lipschitz retractions $X(n)\rightarrow X(n-1)$, $n\geq
2$, exist for $X$ in more general classes of Banach spaces.
