We introduce a new non-negative matrix factorization (NMF) method for ordinal data, called OrdNMF. Ordinal data are categorical data which exhibit a natural ordering between the categories. In particular, they can be found in recommender systems, either with explicit data (such as ratings) or implicit data (such as quantized play counts). OrdNMF is a probabilistic latent factor model that generalizes Bernoulli-Poisson factorization (BePoF) and Poisson factorization (PF) applied to binarized data. Contrary to these methods, OrdNMF circumvents binarization and can exploit a more informative representation of the data. We design an efficient variational algorithm based on a suitable model augmentation and related to variational PF. In particular, our algorithm preserves the scalability of PF and can be applied to huge sparse datasets. We report recommendation experiments on explicit and implicit datasets, and show that OrdNMF outperforms BePoF and PF applied to binarized data.