Integrated satellite terrestrial networks (ISTNs) are becoming a hot research topic in recent years due to the capabilities of high quality, high throughput, and seamless coverage. However, the limited spectrum resources are difficult to meet the needs of a large number of users in ISTNs, and the obstacles and shadowing will seriously affect the communication quality of networks. In this regard, cooperative transmission of non-orthogonal multiple access (NOMA)-assisted ISTNs with the help of multiple terrestrial relays are established to enhance the spectrum efficiency and enlarge the transmission coverage. In this paper, we investigate the performance of NOMA-based ISTNs with relay selection and imperfect successive interference cancellation (SIC). Particularly, the partial relay selection (PRS) scheme is used in the considered system to reach the balance of system performance and complexity. Owing to practical constraints, the imperfect SIC is analyzed for the networks. Based on the PRS scheme and imperfect SIC, we obtain the closed-form expressions for the outage probability (OP) and ergodic capacity (EC) of the considered networks. Besides, to get further insights of key system parameters, the asymptotic analysis for the OP is also derived. Finally, numerical and simulation results are presented to validate the correctness of our analytical results. INDEX TERMS Integrated satellite terrestrial networks (ISTNs), non-orthogonal multiple access (NO-MA), relay selection, successive interference cancellation (SIC).