Lightness perception is a long-standing topic in research on human vision, but very few image-computable models of lightness have been formulated. Recent work in computer vision has used artifical neural networks and deep learning to estimate surface reflectance and other intrinsic image properties. Here we investigate whether such networks are useful as models of human lightness perception. We train a standard deep learning architecture on a novel image set that consists of simple geometric objects with a few different surface reflectance patterns. We find that the model performs well on this image set, generalizes well across small variations, and outperforms three other computational models. The network has partial lightness constancy, much like human observers, in that illumination changes have a systematic but moderate effect on its reflectance estimates. However, the network generalizes poorly beyond the type of images in its training set: it fails on a lightness matching task with unfamiliar stimuli, and does not account for several lightness illusions experienced by human observers.