The continuous new XLindley distribution was introduced by Nawel et al. (2023) as a special case of the polynomial exponential distribution proposed by Beghriche et al. (2022).The current paper introduces the one-parameter discrete analogue distribution of the newXLindley model and studies its main statistical properties. In particular, closed-formexpressions are provided for the moment generating function, mean, variance, quantilefunction, hazard rate function and mean residual life. Moreover, the new distribution hasdiscrete increasing failure rate and both overdispersed and underdispersed count data canbe handled. The estimation of the unknown parameter can be performed by the maximum likelihood method and a Monte Carlo simulation study reveals that this methodprovides satisfactory estimates. Additionally, a first-order integer-valued autoregressiveprocess is constructed from the discrete distribution and, via a simulation study, the conditional maximum likelihood method is recommended for estimation purposes. In orderto assess the usefulness in practical applications, the proposed distribution and the associated first-order autoregressive process are compared to other competing distributionsand processes, using to this end several real data sets. In the context of statistical qualitycontrol, finally a cumulative sum control chart is developed for monitoring the processmean. To illustrate its usefulness, both simulation and real data analysis are performed.