  In this paper we study the asymptotic behavior of solutions to the
subelliptic $p$-Poisson equation as $p\to +\infty$ in Carnot Carath\'eodory
spaces. In particular, introducing a suitable notion of differentiability, we
extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi [Rend.
Sem. Mat. Univ. Politec. Torino, 1989, Special Issue, 15-68] and we prove that
limits of such solutions solve in the sense of viscosity a hybrid first and
second order PDE involving the $\infty-$Laplacian and the Eikonal equation.
