Accurately predicting the horizontal component of the ground magnetic field perturbation (dBH), which can be used to calculate the Geomagnetically Induced Currents (GICs), is crucial for estimating the space weather impact of geomagnetic disturbances. In this work, we develop a new data-driven model GeoDGP using deep Gaussian process (DGP), which is a Bayesian non-parametric approach. The model provides global probabilistic forecasts of dBH at 1-minute time cadence and with arbitrary spatial resolutions. We evaluate the model comprehensively on a wide range of geomagnetic storms, including the 2024 Gannon extreme storm. The results show that GeoDGP significantly outperforms the state-of-the-art physics-based first-principles Space Weather Modeling Framework (SWMF) Michigan Geospace model and the data-driven DAGGER model.