  We study the image of $\ell$-adic representations attached to subvarieties of
Shimura varieties $Sh_K(G,X)$ that are not contained in a smaller Shimura
subvariety and have no isotrivial components. We show that, for $\ell$ large
enough (depending on the Shimura datum $(G,X)$ and the subvariety), such image
contains the $\mathbb{Z}_\ell$-points coming from the simply connected cover of
the derived subgroup of $G$. This can be regarded as a geometric version of the
integral $\ell$-adic Mumford-Tate conjecture.
