  We report on a non-perturbative study of two dimensional $\cN=(2,2)$ super
QCD. Our lattice formulation retains a single exact supersymmetry at non-zero
lattice spacing, and contains $N_f$ fermions in the fundamental representation
of a $U(N_c)$ gauge group. The lattice action we employ contains an additional
Fayet-Iliopoulos term which is also invariant under the exact lattice
supersymmetry. This work constitutes the first numerical study of this theory
which serves as a toy model for understanding some of the issues that are
expected to arise in four dimensional super QCD. We present evidence that the
exact supersymmetry breaks spontaneously when $N_f<N_c$ in agreement with
theoretical expectations.
