Fully distributed hydrological models take into account the spatial variability of a catchment, and allow for assessing its hydrological response at virtually any location. However, these models can be time-consuming when it comes to model runtime and calibration, especially for large-scale catchments. Meanwhile, deep learning models have shown great potential in the field of hydrological modelling, but a multivariable, fully distributed hydrological deep learning model is still lacking. To address the aforementioned challenges associated with fully distributed models and deep learning models, we explore the possibility of developing a fully distributed multivariable deep learning model by using Graph Neural Networks (GNN), an extension of deep learning methods to non-Euclidean topologies. We develop a surrogate model of wflow_sbm, a fully distributed, physics-based hydrological model, by exploiting the similarities between its underlying functioning and GNNs. The GNN model uses the same input as wflow_sbm: gridded static parameters based on physical characteristics of the catchment and gridded dynamic meteorological forcings. The GNN model is trained to approximate wflow_sbm outputs, consisting of multiple gridded hydrological variables such as streamflow, actual evapotranspiration, subsurface flow, saturated and unsaturated groundwater storage, snow storage, and runoff. Our results show that the GNN model accurately predicts multiple hydrological variables in unseen catchments (median KGE=0.76), and can serve as an emulator of wflow_sbm with a shorter runtime. We furthermore demonstrate how the GNN model can function up to a prediction horizon of a full year, using physical system states to account for system memory, as well as a curriculum learning strategy combined with a multi-step ahead loss function during training. Overall, this study contributes to the field of fully distributed modelling using a deep learning approach.