In light of your most recent edit, the answer is now "yes" by Fodor's Lemma: any regressive function $f$ is constant on a stationary, and hence cofinal, set; since you now demand $f$ to be nondecreasing, this means that $f$ attains a maximum value $\alpha_f$. Mapping every $f$ to the constant function $d_f: \gamma\mapsto\alpha_f+1$ then gives an example of your desired map.
EDIT: Using Fodor's Lemma is unnecessary. Here's a non-nuclear proof:
As noted above, it's enough to show that the range of each such $f$ is bounded in $\omega_1$. Suppose $g$ were a regressive, non-decreasing function with unbounded range. Define now a sequence $\alpha_i$, with $\alpha_0=1$ and $\alpha_{n+1}$ the least $\alpha$ such that $g(\alpha)>\alpha_n$; note that by the assumption that $g$ has unbounded range, such a sequence exists.
Now let $\beta=\lim \alpha_i$. Since $g$ is nondecreasing, $g(\beta)\ge\beta$, contradicting the assumption that $g$ is regressive. QED