We propose and investigate a compressive architecture for estimation and tracking of sparse spatial channels in millimeter (mm) wave picocellular networks. The base stations are equipped with antenna arrays with a large number of elements (which can fit within compact form factors because of the small carrier wavelength) and employ radio frequency (RF) beamforming, so that standard least squares adaptation techniques (which require access to individual antenna elements) are not applicable. We focus on the downlink, and show that "compressive beacons," transmitted using pseudorandom phase settings at the base station array, and compressively processed using pseudorandom phase settings at the mobile array, provide information sufficient for accurate estimation of the two-dimensional (2D) spatial frequencies associated with the directions of departure of the dominant rays from the base station, and the associated complex gains. This compressive approach is compatible with coarse phase-only control, and is based on a near-optimal sequential algorithm for frequency estimation which can exploit the geometric continuity of the channel across successive beaconing intervals to reduce the overhead to less than 1% even for very large (32 x 32) arrays. Compressive beaconing is essentially omnidirectional, and hence does not enjoy the SNR and spatial reuse benefits of beamforming obtained during data transmission. We therefore discuss system level design considerations for ensuring that the beacon SNR is sufficient for accurate channel estimation, and that inter-cell beacon interference is controlled by an appropriate reuse scheme.