In this paper we propose a new stationary first-order non-negative integer valued autoregressive process with geometric marginals based on a generalised version of the negative binomial thinning operator. In this manner we obtain another process that we refer to as a generalised stationary integer-valued autoregressive process of the first order with geometric marginals. This new process will enable one to tackle the problem of overdispersion inherent in the analysis of integer-valued time series data, and contains the new geometric process as a particular case. In addition various properties of the new process, such as conditional distribution, autocorrelation structure and innovation structure, are derived. We discuss conditional maximum likelihood estimation of the model parameters. We evaluate the performance of the conditional maximum likelihood estimators by a Monte Carlo study. The proposed process is fitted to time series of number of weekly sales (economics) and weekly number of syphilis cases (medicine) illustrating its capabilities in challenging cases of highly overdispersed count data.