Cellular networks of massive MIMO base-stations employing TDD/OFDM and relying on uplink training for both downlink and uplink transmission are viewed as an attractive candidate for 5G deployments, as they promise high area spectral and energy efficiencies with relatively simple low-latency operation. We investigate the use of non-orthogonal uplink pilot designs as a means for improving the area spectral efficiency in the downlink of such massive MIMO cellular networks. We develop a class of pilot designs that are locally orthogonal within each cell, while maintaining low inner-product properties between codes in different cells. Using channel estimates provided by observations on these codes, each cell independently serves its locally active users with MU-MIMO transmission that is also designed to mitigate interference to a subset of `strongly interfered' out-of-cell users. As our simulation-based analysis shows, such cellular operation based on the proposed codes yields user-rate CDF improvement with respect to conventional operation, which can be exploited to improve cell and/or cell-throughput performance.