Geographically weighted small area methods have been studied in literature for small area estimation. Although these approaches are useful for the estimation of small area means efficiently under strict parametric assumptions, they can be very sensitive to outliers in the data. In this paper, we propose a robust extension of the geographically weighted empirical best linear unbiased predictor (GWEBLUP). In particular, we introduce robust projective and predictive small area estimators under spatial non-stationarity. Mean squared error estimation is performed by two different analytic approaches that account for the spatial structure in the data. The results from the model-based simulations indicate that the proposed approach may lead to gains in terms of efficiency. Finally, the methodology is demonstrated in an illustrative application for estimating the average total cash costs for farms in Australia.