This paper considers inference for both spatial lattice data with possibly irregularly shaped sampling region and non-lattice data, by extending the recently proposed self-normalization (SN) approach from stationary time series to the spatial setup. A nice feature of the SN method is that it avoids the choice of tuning parameters, which are usually required for other non-parametric inference approaches. The extension is non-trivial as spatial data has no natural one-directional time ordering. The SN-based inference is convenient to implement and is shown through simulation studies to provide more accurate coverage compared with the widely used subsampling approach. We also illustrate the idea of SN using a real data example.