We study the channel estimation problem for a reconfigurable intelligent surface (RIS)-assisted millimeter-wave (mmWave) multi-user multiple-input multiple-output (MU-MIMO) system. In particular, it is assumed that the channel between a RIS and a base station (BS) exhibits a near-field line-of-sight (LoS) channel, which is a dominant signal path in mmWave communication systems. Due to the high-rankness and non-sparsity of the RIS-BS channel matrix in our system, the state-of-the-art (SOTA) methods, which are constructed based on far-field or near-field non-LoS (NLoS) channel, cannot provide attractive estimation performances. We for the first time propose an efficient near-field LoS/NLoS channel estimation method for RIS-assisted MU-MIMO systems by means of a piece-wise low-rank approximation. Specifically, an effective channel (to be estimated) is partitioned into piece-wise effective channels containing low-rank structures and then, they are estimated via collaborative low-rank approximation. The proposed method is named PW-CLRA. Via simulations, we verify the effectiveness of the proposed PW-CLRA.