Emerging wireless communication systems at millimeter-wave (mmW) require wideband phased-arrays transceivers not only for high data-rate communications but also for fast direction finding using beam training algorithms. In real world, scanning both azimuth and elevation is also important for accurate localization over three-dimensional (3D) space. This brief presents the system analysis and demonstration of frequency dispersive 3D beam training algorithm using a $2 \times 2$ planar array integrated in a single chip suited for low-latency mmW wireless communications. System-level issues with high search latency in earlier time-division based beam training algorithms and the need for multiple ICs for 3D beam training are addressed with a large delay range true-time-delay (TTD) based spatial signal processor (SSP) together with the frequency-dependent rainbow beam training algorithm. Trade-offs between angular coverage efficiency over the 3D space and required hardware delay range are analyzed. Measured results on the $2 \times 2$ antenna array demonstrate the efficacy of the 3D beam training algorithm achieving 50% spherical coverage efficiency realized with the 3.75ns IC delay range over 800MHz bandwidth.