Starting from any linear pseudo-Anosov map $\varphi$ on a surface of genus $g \geqslant 2$, we construct explicitly a family of non-linear pseudo-Anosov maps $f$ by adapting the construction of Smale's Derived from Anosov maps on the two-torus. This is done by perturbing $\varphi$ at some fixed points. We first consider perturbations at every conical fixed point and then at regular fixed points. We establish the existence of a measure $\mu$, supported by the non-trivial unique minimal component of the stable foliation of $f$, with respect to which $f$ is mixing. In the process, we construct a uniquely ergodic generalized interval exchange transformation with a wandering interval that is semi-conjugated to a self-similar interval exchange transformation. This generalized interval exchange transformation is obtained as the Poincar\'e map of a flow renormalized by $f$. When $f$ is $\mathcal{C}^2$, the flow and the generalized interval exchange transformation are $\mathcal{C}^1$.