  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 $\mathcal{C}^1$.
