Let $S$ be a unit sphere in the [Urysohn space](https://en.wikipedia.org/wiki/Urysohn_universal_space) $\mathbb{U}$.
Is it true that any isometry $S\to S$ can be extended to an isometry $\mathbb{U}\to \mathbb{U}$?



Sphere in Urysohn space
This is not true, no.
There is a proof in Section 4.4 of this [old paper of mine](http://math.univ-lyon1.fr/%7Emelleray/NoteBeerSheva3.pdf) ; the key fact is that if $B$ is an open unit ball in the Urysohn space $\mathbb U$, then $\mathbb U$ is isometric to $\mathbb U \setminus B$.


(the proof of the fact about extension of isometries is given for a ball in the paper, but the same argument works if one replaces "ball" by "sphere" in the proof)


