Astronomical observations of extended sources, such as cubes of integral field spectroscopy (IFS), encode auto-correlated spatial structures that cannot be optimally exploited by standard methodologies. This work introduces a novel technique to model IFS datasets, which treats the observed galaxy properties as realizations of an unobserved Gaussian Markov random field. The method is computationally efficient, resilient to the presence of low-signal-to-noise regions, and uses an alternative to Markov Chain Monte Carlo for fast Bayesian inference, the Integrated Nested Laplace Approximation (INLA). As a case study, we analyse 721 IFS data cubes of nearby galaxies from the CALIFA and PISCO surveys, for which we retrieve the maps of the following physical properties: age, metallicity, mass and extinction. The proposed Bayesian approach, built on a generative representation of the galaxy properties, enables the creation of synthetic images, recovery of areas with bad pixels, and an increased power to detect structures in datasets subject to substantial noise and/or sparsity of sampling. A snippet code to reproduce the analysis of this paper is available in the COIN toolbox, together with the field reconstructions of the CALIFA and PISCO samples.
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