The mutual information on a multiple-input-multiple-output (MIMO) link between a transmitter-receiver pair in a wireless network with co-channel interferers at correlated spatial positions is analyzed when the number of receiver antennas is large and the rank of transmit covariance matrices are finite. Asymptotic expressions for the mutual information are derived as a function of the rank of the transmit covariance matrices, number of receiver antennas, density of transmitters, and path-loss exponent. The results are applied to hard-core networks and the uplink of massive MIMO cellular networks. For the latter model, an approximation to the cumulative-distribution-function of a randomly located link in a given cell is provided. The results provide insight into achievable data rates in MIMO networks with users at correlated spatial positions.