Source separation, which consists in decomposing data into meaningful structured components, is an active research topic in many areas, such as music and image signal processing, applied physics and text mining. In this paper, we introduce the Positive $α$-stable (P$α$S) distributions to model the latent sources, which are a subclass of the stable distributions family. They notably permit us to model random variables that are both nonnegative and impulsive. Considering the Lévy distribution, the only P$α$S distribution whose density is tractable, we propose a mixture model called Lévy Nonnegative Matrix Factorization (Lévy NMF). This model accounts for low-rank structures in nonnegative data that possibly has high variability or is corrupted by very adverse noise. The model parameters are estimated in a maximum-likelihood sense. We also derive an estimator of the sources given the parameters, which extends the validity of the generalized Wiener filtering to the P$α$S case. Experiments on synthetic data show that Lévy NMF compares favorably with state-of-the art techniques in terms of robustness to impulsive noise. The analysis of two types of realistic signals is also considered: musical spectrograms and fluorescence spectra of chemical species. The results highlight the potential of the Lévy NMF model for decomposing nonnegative data.