This is kind of a continuation of a recent (closed) question.

Is there an order-preserving surjective function $f:\omega^\omega \to [0,\infty)$ (where for $a,b\in \omega^\omega$ we have $a\leq b$ if $a(n) \le b(n)$ for all $n\in \omega$)?

Thanks to Jeremy Rickard who made me aware that a previous version of this question was trivial and therefore uninteresting.