Small area inference based on mixed models, i.e. models that contain both fixed and random effects, are the industry standard for this field, allowing between area heterogeneity to be represented by random area effects. Use of the linear mixed model is ubiquitous in this context, with maximum likelihood, or its close relative, REML, the standard method for estimating the parameters of this model. These parameter estimates, and in particular the resulting predicted values of the random area effects, are then used to construct empirical best linear unbiased predictors (EBLUPs) of the unknown small area means. It is now well known that the EBLUP can be unstable when there are outliers in sample data, and an outlier-robust EBLUP, or REBLUP, has been proposed by Sinha and Rao (2009), based on modifying the parameter estimating functions to make them less sensitive to sample outliers. Unfortunately, these modified estimating functions can be numerically unstable, and mean squared error estimation for the REBLUP is not straightforward. Taking a somewhat different approach, Chambers and Mokhtarian (2013) proposed an outlier robust block bootstrap approach to fitting a linear mixed model in the presence of both area level and unit level outliers. A natural extension of this bounded block bootstrap can then be used to define an outlier robust version of the EBLUP and a simple way of estimating its mean squared error. This approach is described in this paper, together with simulation results that provides some evidence for our claim that the new method is robust to the influence of outliers. In particular, it leads to an easily computed version of the REBLUP and an easily computed and stable estimate of its mean squared error.