Let $A$ and $B$ be CW complexes and denote by $\operatorname{map}(A,B)$ their mapping space. Suppose that both $A$ and $B$ are connected, that both spaces have finitely many nontrivial homotopy groups and that all homotopy groups of $A$ and $B$ are finite. I am trying to show that $\operatorname{map}(A,B)$ then has only finitely many path components, i.e. that the set of homotopy classes of maps between $A$ and $B$ is finite.
This is implicitly claimed in a paper that I read, but I am unable to find an argument for this. I would be grateful for any help in that regard.