We present the first high-resolution map of the cold molecular gas distribution, as traced by CO(2-1) emission with ALMA, in a prominent ram pressure stripped tail. The Norma cluster galaxy ESO 137-001 is undergoing a strong interaction with the surrounding intra-cluster medium and is one of the nearest jellyfish galaxies with a long multi-phase tail. We have mapped the full extent of the tail at 1" (350 pc) angular resolution and found a rich distribution of mostly compact CO regions extending to nearly 60 kpc in length and 25 kpc in width. In total about 10^9 M_sun of molecular gas was detected. The CO features are found predominantly at the heads of numerous small-scale (~ 1.5 kpc) fireballs (i.e., star-forming clouds with linear streams of young stars extending toward the galaxy) but also of large-scale (~ 8 kpc) super-fireballs, and double-sided fireballs that have additional diffuse ionized gas tails extending in the direction opposite to the stellar tails. The new data help to shed light on the origin of the molecular tail - CO filaments oriented in the direction of the tail with only diffuse associated Halpha emission are likely young molecular features formed in situ, whereas other large CO features tilted with respect to the tail may have originated from the densest gas complexes that were pushed gradually away from the disk. The ALMA observations of ESO 137-001, together with observations from HST, Chandra and VLT/MUSE, offer the most complete view of a spectacular ram pressure stripped tail to date.