I think I should be posting this as a comment, but as i don't have enough reputation to comment I take the liberty to post this as an answer.
I am here just elaborating on Eric Wofsey's answer.
Let's take the case, say, $X=\mathbb{N}$ and $Y=I=[0,1]$. Then $Map(X,Y)$ is Hilbert cube.
On the other hand, the definition of a CW-complex says that a CW complex has to be the union of finite skelta, and for reasons explained in Eric Wofsey's answer, this is not the case.