The Cheeger-differentiability of maps $f$ belonging to Hajlasz spaces $M^{1,p}(X,E)$ is proved. The assumption are rather simple. The metric space $X$ is assumed to support a doubling measure $\mu $ for which a Poincaré inequality is satisfied. The target space $E$ is assumed to be a RNP Banach space. Finally, the parameter $p$ must be large enough compared to the doubling constant of the measure $\mu $ (a precise and optimal dependance is given by the way). Notice that the of Banach spaces considered is the largest possible (it gives a new characterization of RNP in term of Cheeger-differentiability).