The unprecedented scale and sophistication of wind turbine technologies call for wind forecasts of high spatial resolution, i.e. turbine-tailored forecasts, to inform several operational decisions at the turbine level. Towards that, this paper is concerned with leveraging the hub-height measurements collected from a fleet of turbines on a farm to make turbine-specific short-term wind speed and power predictions. We find that the wind propagation across a dense grid of turbines induces strong spatial and temporal dependencies in the within-farm wind field, but also gives rise to high-frequency high-magnitude fluctuations which may compromise the predictive accuracy of several data-driven forecasting methods. To capture both aspects, we propose to model the total variability in the within-farm wind speed field as a combination of two independent stochastic process terms. The first term reconstructs and extrapolates the wind speed field by learning the complex spatio-temporal dependence structure using hub-height turbine-level data. The second term accounts for high-frequency high-magnitude fluctuations that are not informed by near-term spatio-temporal dependencies. The two terms are coupled to make probabilistic wind speed forecasts at each turbine, which are then translated into turbine-specific power predictions via wind power curves. Evaluation on more than 3,000,000 data points from a wind farm dataset provides a strong empirical evidence in favor of the proposed method’s forecasting accuracy. On average, our proposed method achieves 9% accuracy improvement relative to persistence forecasts, and 7–9% relative to a set of widely recognized forecasting methods such as autoregressive-based models and Gaussian Processes.