Summary. Remote sensing is one example where data sets that vary across space and time have become so large that ‘standard’ approaches employed by statistical modellers for applied analysis are no longer feasible. We present a Bayesian methodology, which makes use of recently developed algorithms in applied mathematics, for the analysis of large space–time data sets. In particular, a Markov chain Monte Carlo algorithm is proposed for the efficient estimation of spatial dynamic factor models. The spatial dynamic factor model is specified whereby spatial dependence is modelled though the columns of the factor loadings matrix by using a Gaussian Markov random field. Krylov subspace methods are used to take advantage of the sparse matrix structures that are inherent in the model. The methodology is used to analyse remotely sensed data from the Moderate Imaging Spectroradiometer satellite. In particular, the methodology proposed is used in conjunction with high resolution imagery for the classification, in terms of land type, of two regions in central Queensland, Australia.