Beamspace channel estimation in millimeter-wave (mmWave) massive MIMO system is a very challenging task, especially when the number of radio-frequency chains is limited. To address this problem, we present a novel approximate message passing (AMP)-based network with deep residual learning, referred to as LampResNet. It mainly consists of two components: 1) a learned AMP (LAMP) network and 2) a deep residual learning network (ResNet). The former utilizes the sparsity property of beamspace channel matrix and is employed to obtain a preliminary estimation result, while the latter is designed to reduce the impact of channel noise and further refine the coarse estimation obtained by the LAMP network. Simulation results validate the efficiency of the proposed network.