Reconfigurable intelligent surface (RIS) is a promising technique to enhance the performance of physical-layer key generation (PKG) due to its ability to smartly customize the radio environments. Existing RIS-assisted PKG methods are mainly based on the idealistic assumption of an independent and identically distributed (i.i.d.) channel model at both the transmitter and the RIS. However, the i.i.d. model is inaccurate for a typical RIS in an isotropic scattering environment. Also, neglecting the existence of channel spatial correlation would degrade the PKG performance. In this paper, we establish a general spatially correlated channel model in multi-antenna systems and propose a new PKG framework based on the transmit and the reflective beamforming at the base station (BS) and the RIS. Specifically, we derive a closed-form expression for characterizing the key generation rate (KGR) and obtain a globally optimal solution of the beamformers to maximize the KGR. Furthermore, we analyze the KGR performance difference between the one adopting the assumption of the i.i.d. model and that of the spatially correlated model. It is found that the beamforming designed for the correlated model outperforms that for the i.i.d. model while the KGR gain increases with the channel correlation. Simulation results show that compared to existing methods based on the i.i.d. fading model, our proposed method achieves about $5$ dB performance gain when the BS antenna correlation $ρ$ is $0.3$ and the RIS element spacing is half of the wavelength.