  We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg
to the setting of singular integrals with operator-valued kernel that act on
vector-valued functions. Here, `vector-valued' means `taking values in a
function lattice with the UMD (unconditional martingale differences) property'.
A similar extension (but for general UMD spaces rather than UMD lattices) of
Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier
by the first author, and it has found applications in the work of Mayboroda and
Volberg on square-functions and rectifiability. Our local version requires
several elaborations of the previous techniques, and raises new questions about
the limits of the vector-valued theory.
