In this article a geometrically distributed integer-valued autoregressive model of order one based on the mixed thinning operator is introduced. This new thinning operator is defined as a probability mixture of two well known thinning operators, binomial and negative binomial thinning. Some model properties are discussed. Method of moments and the conditional least squares are considered as possible approaches in model parameter estimation. Asymptotic characterization of the obtained parameter estimators is presented. The adequacy of the introduced model is verified by its application on a certain kind of real-life counting data, while its performance is evaluated by comparison with two other INAR(1) models that can be also used over the observed data.