A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.

An auto-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we have $d(x,y) = d(\iota(x), \iota(y))$.

Does there exist a bounded metric space with a non-surjective auto-isometry?

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.

A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we have $d(x,y) = d(\iota(x), \iota(y))$.

Does there exist a bounded metric space with a non-surjective self-isometry?