We consider a two-stage unit-level model for small areas with continuous survey responses. Typically survey data have responses with outliers, gaps and ties, and the distributions of the responses might be skewed. Therefore, predictive inference using a hierarchical Bayesian model with normality at both levels (responses and random effects) might not be robust against these features. So we provide a two-level non-parametric Bayesian model with a Dirichlet process at each stage, thereby permitting a more robust predictive inference. We show how to fit the four versions of this model (e.g., one version has Dirichlet processes at both levels) using Markov chain Monte Carlo methods. An application on body mass index and a simulation study are discussed to compare the four models. While it is difficult to tell which model is preferred, one might want to use the model with Dirichlet processes at both levels (it robustifies both levels against non-normality).