Reconfigurable intelligent surfaces (RISs) have attracted extensive attention from industry and academia. In RIS-assisted wireless communication systems, practical channel measurements and modeling serve as the foundation for system design, network optimization, and performance evaluation. In this paper, a RIS time-domain channel measurement system, based on a software defined radio platform, is developed for the first time to investigate the small-scale fading characteristics of RIS-assisted channels. We present RIS channel measurements in corridor and laboratory scenarios and compare the power delay profile of the channel without RIS, with RIS specular reflection, and with RIS intelligent reflection. The multipath component parameters and cluster parameters based on the Saleh-Valenzuela model are extracted. We find that the power delay profiles (PDPs) of the RIS-assisted channel fit the power-law decay model better than the common exponential decay model and approximate the law of square decay. Through intelligent reflection, the RIS can decrease the delay and concentrate the energy of the virtual line-of-sight (VLoS) path, thereby reducing the delay spread and mitigating multipath fading. Furthermore, the cluster characteristics of RIS-assisted channels are highly dependent on the measurement environment. In the laboratory scenario, a single cluster dominated by the VLoS path with smooth envelope is observed. On the other hand, in the corridor scenario, some additional clusters introduced by the RIS reflection are created.