  We proceed with studying the q-analogues of Cartan domains introduced in
q-alg/9703005 and turn to the case of a ball in the space of complex matrices.
An explicit expression for a positive $U_q\frak{su}_{nm}$-invariant integral
(see math.QA/9803110) is used to define q-analogues for weighted Bergman
spaces. The orthogonal projections onto these spaces are integral operators;
this work presents an explicit formula for their kernels.
