Photometric redshifts are necessary for enabling large-scale multicolour galaxy surveys to interpret their data and constrain cosmological parameters. While the increased depth of future surveys such as the Large Synoptic Survey Telescope (LSST) will produce higher precision constraints, it will also increase the fraction of sources that are blended. In this paper, we present a Bayesian photometric redshift (BPZ) method for blended sources with an arbitrary number of intrinsic components. This method generalizes existing template-based BPZ methods, and produces joint posterior distributions for the component redshifts that allow uncertainties to be propagated in a principled way. Using Bayesian model comparison, we infer the probability that a source is blended and the number of components that it contains. We extend our formalism to the case where sources are blended in some bands and resolved in others. Applying this to the combination of LSST- and Euclid-like surveys, we find that the addition of resolved photometry results in a significant improvement in the reduction of outliers over the fully blended case. We make available blendz, a Python implementation of our method.