Calibration of model parameters based on full-field measurements is now very commonly employed in mechanical and material engineering practices. This may be done in different ways, for instance by using advanced frameworks such as the Virtual Field Method, or Integrated Digital Image Correlation (I-DIC). In the present paper, we concentrate on a classical paradigm where full displacement fields are recovered using generic DIC algorithms, and model parameters are subsequently identified using FEMU by minimising a measure of the difference between the simulated displacement fields and that obtained by DIC. Usually, the model parameters to be identified are constant across the specimen. However, in this study, we aim to account for macroscopically observable variations in material properties and seek to determine material parameters in the form of unknown spatial fields. This introduces several challenges, amongst which (i) appropriately discretising the field of unknown parameters, (ii) computing gradients inverse problems’ cost function with respect to an arbitrary number of parameters and (iii) using a well-crafted regularisation methodology to make the inverse problem well-posed. Points one and two will be addressed by means of finite element discretisations and adjoints-based methodologies, respectively, the second point being made very efficient using the automatic differentiation capabilities of the FEniCS software. For the third point, we develop a stochastic-PDE-based regularisation technique. This technique may be viewed as using a Matérn Gaussian Process prior in a Bayesian inverse problem setting, with the distinct advantage over standard, Kernel-based Gaussian Process priors that only sparse operator inversions are required. We will demonstrate that the approach efficiently delivers correct fields of material constants when associated with algorithms to automatically adjust the regularisation hyperparameters in an empirical Bayes fashion.