Sea surface temperature (SST) variability plays a key role in the global weather and climate system, with phenomena such as El Niño-Southern Oscillation (ENSO) regarded as a major source of interannual climate variability at the global scale. The ability to make long-range forecasts of SST variations and extreme marine heatwave events have potentially significant economic and societal benefits, especially in a warming climate. We have developed a deep learning time series prediction model (Unet-LSTM), based on more than 70 years (1950–2021) of ECMWF ERA5 monthly mean SST and 2-m air temperature data, to predict global 2-dimensional SSTs up to a 24-month lead. Model prediction skills are high in the equatorial and subtropical Pacific. We have assessed the ability of the model to predict SST anomalies in the Niño3.4 region, an ENSO index in the equatorial Pacific, and the Blob marine heatwave events in the northeast Pacific in detail. An assessment of the predictions of the 2019–2020 El Niño and the 2016–2017 and 2017–2018 La Niña show that the model has skill up to 18 months in advance. The prediction of the 2015–2016 extreme El Niño is less satisfactory, which suggests that subsurface ocean information may be crucial for the evolution of this event. Note that the model makes predictions of the 2-d monthly SST field and Nino 3.4 is just one region embedded in the global field. The model also shows long lead prediction skills for the northeast Pacific marine heatwave, the Blob. However, the prediction of the marine heatwaves in the southeast Indian Ocean, the Ningaloo Niño, shows a short lead prediction. These results indicate the significant potential of data-driven methods to yield long-range predictions of SST anomalies.