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A three-parameter lifetime distribution is proposed, named extended Chen-Poisson distribution, by compounding the Chen and zero-truncated Poisson distributions. The new distribution belongs to the unified Poisson family, where both distributions of the minimum and maximum are merged into one. Several properties of the distribution are studied. The proposed distribution is quite flexible since it accommodates different complex hazard shapes. Inference is based on the maximum likelihood method in the presence of a right-censoring mechanism. A simulation study is performed to evaluate the properties of the parameters estimators. Two real lifetime data sets are analysed for purposes of comparison with other generalizations of the Chen distribution, as well as with other members of the unified Poisson family. The obtained results allow to highlight the potential of the new distribution.