Channel estimation (CE) for millimeter-wave (mmWave) lens-array suffers from prohibitive training overhead, whereas the state-of-the-art solutions require an extra complicated radio frequency phase shift network. By contrast, lens-array using antenna switching network (ASN) simplifies the hardware, but the associated CE is a challenging task due to the constraint imposed by ASN. This paper proposes a compressive sensing (CS)-based CE solution for full-dimensional (FD) lens-array, where the mmWave channel sparsity is exploited. Specifically, we first propose an approach of pilot training under the more severe haraware constraint imposed by ASN, and formulate the associated CE of lens-array as a CS problem. Then, a redundant dictionary is tailored for FD lens-array to combat the power leakage caused by the continuous angles of multipath components. Further, we design the baseband pilot signals to minimize the total mutual coherence of the measurement matrix based on CS theory for more reliable CE performance. Our solution provides a framework for applying CS techniques to lens-array using simple and practical ASN. Simulation results demonstrate the effectiveness of the proposed scheme.