Many studies have considered mixed Poisson distributions as alternatives for fitting count data with overdispersion. However, in some cases, the data have an abundance of zeros and ones which makes modelling using distributions with no consideration to the inflated values less desirable. This study aims to introduce a new distribution for count data with inflated values at zero and one, known as zero–one-inflated Poisson–Lindley distribution. The statistical properties of the proposed distribution are discussed. Furthermore, the maximum likelihood and method of moments for the parameters of the proposed distribution are developed. A simulation study is conducted to investigate the performance of the zero–one-inflated Poisson–Lindley distribution in describing overdispersed data with excess zeros and ones by changing the proportion of zeros and ones in the data. It is found that the fitting of the zero–one-inflated Poisson–Lindley distribution always gives a larger log-likelihood value than the fitting of the zero–one-inflated Poisson distribution. The results from the applications of the real datasets with an overdispersed property as well as a large number of ones and zeros conclude that the proposed distribution provides the best fit compared to other contending distributions in the study.