Sea ice plays an important role in stabilising the Earth system. Yet, representing its dynamics remains a major challenge for models, as the underlying processes are scale-invariant and highly anisotropic. This poses a dilemma: physics-based models that faithfully reproduce the observed dynamics are computationally costly, while efficient AI models sacrifice realism. Here, to resolve this dilemma, we introduce GenSIM, the first generative AI model to predict the evolution of the full Arctic sea-ice state at 12-hour increments. Trained for sub-daily forecasting on 20 years of sea-ice-ocean simulation data, GenSIM makes realistic predictions for 30 years, while reproducing the dynamical properties of sea ice with its leads and ridges and capturing long-term trends in the sea-ice volume. Notably, although solely driven by atmospheric reanalysis, GenSIM implicitly learns hidden signatures of multi-year ice-ocean interaction. Therefore, generative AI can extrapolate from sub-daily forecasts to decadal simulations, while retaining physical consistency.