Detecting phase arrivals and pinpointing the arrival times of seismic phases in seismograms is crucial for many seismological analysis workflows. For land station data, machine learning methods have already found widespread adoption. However, deep learning approaches are not yet commonly applied to ocean bottom data due to a lack of appropriate training data and models. Here, we compiled an extensive and labeled ocean bottom seismometer (OBS) data set from 15 deployments in different tectonic settings, comprising ∼90,000 P and ∼63,000 S manual picks from 13,190 events and 355 stations. We propose PickBlue, an adaptation of the two popular deep learning networks EQTransformer and PhaseNet. PickBlue joint processes three seismometer recordings in conjunction with a hydrophone component and is trained with the waveforms in the new database. The performance is enhanced by employing transfer learning, where initial weights are derived from models trained with land earthquake data. PickBlue significantly outperforms neural networks trained with land stations and models trained without hydrophone data. The model achieves a mean absolute deviation of 0.05 s for P-waves and 0.12 s for S-waves, and we apply the picker on the Hikurangi Ocean Bottom Tremor and Slow Slip OBS deployment offshore New Zealand. We integrate our data set and trained models into SeisBench to enable an easy and direct application in future deployments.