In this paper, we propose a new first-order non-negative integervalued autoregressive [INAR(1)] process with Poisson–geometric marginals based on binomial thinning for modeling integer-valued time series with overdispersion. Also, the new process has, as a particular case, the Poisson INAR(1) and geometric INAR(1) processes. The main properties of the model are derived, such as probability generating function, moments, conditional distribution, higher-order moments, and jumps. Estimators for the parameters of process are proposed, and their asymptotic properties are established. Some numerical results of the estimators are presented with a discussion of the obtained results. Applications to two real data sets are given to show the potentiality of the new process.