Let $\omega$ be the set of natural numbers. $2^\omega$ is the Cantor space.
Suppose $K$, $L \subset 2^\omega$ are compact, and there is an isometry $f: K \to L$. Then how could one extend $f$ to an isometry from $2^\omega$ to $2^\omega$? Here we are considering $2^\omega$ with the minimum difference metric, which gives the standard product topology; i.e.
$ d(x,y) = 2^{-\min \{ n : x(n) \neq y(n) \}}. $