Missing values are ubiquitous in (data) science, with potential detrimental consequences for any statistical analysis. As a consequence, a wealth of methods and theoretical results have been developed in recent years. Still, many questions remain open, in particular in the case of general non-monotone missing at random (MAR). In this work, we extend nonparametric Bayesian theory to this MAR setting. We introduce a general theorem of posterior contraction under MAR and an additional mild positivity condition. Using this result, we are able to show that, despite the missing values, the density of the uncontaminated data can be estimated with the minimax posterior contraction rate up to log factors. To the best of our knowledge, this is the first nonparametric result showing that the uncontaminated distribution can be consistently estimated under Rubin's MAR definition. As a consequence, we obtain an algorithm that takes data contaminated with missing values and returns a sample from a provably consistent estimate of the uncontaminated distribution.