Where can I find proof of the following classical fact?
The existence of a nontrivial Killing vector field on a compact Riemannian manifold $M$ is equivalent to the existence of a nontrivial $\Bbb S^1$-action on $M$.
Is there any counterexample in non-compact case?
The group $\text{Isom}(M)$ has the form of a Lie group; if $M$ is compact, this Lie group is compact.
Given any non-trivial Killing vector field, this gives a map $X: \Bbb R \to \text{Isom}(M)$; its image is a commutative subgroup. The closure of the image of $X$ is still commutative (the equation $ab = ba$ is true on an open dense subset of $\bar X \times \bar X$) and still a subgroup. If $M$ is compact, this subgroup must then be compact; because $\bar X$ is a compact, connected, non-trivial abelian Lie group we thus have $\bar X \cong T^n$ for some $n>0$. In particular, there is a circle subgroup of $\text{Isom}(M)$, and hence a faithful action of $S^1$ on $M$ by isometries.