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.