© 2016 The Authors. We present ALMA observations of the CO(1-0) and CO(3-2) line emission tracing filaments of cold molecular gas in the central galaxy of the cluster PKS 0745-191. The total molecular gas mass of 4.6 ± 0.3 × 109 M⊙, assuming a Galactic XCO factor, is divided roughly equally between three filaments each extending radially 3-5 kpc from the galaxy centre. The emission peak is located in the SE filament ~ 1 arcsec (2 kpc) from the nucleus. The velocities of the molecular clouds in the filaments are low, lying within ± 100 km s-1 of the galaxy's systemic velocity. Their full width at half-maximum (FWHM) are less than 150 km s-1 which is significantly below the stellar velocity dispersion. Although the molecular mass of each filament is comparable to a rich spiral galaxy, such low velocities show that the filaments are transient and the clouds would disperse on <107 yr time-scales unless supported, likely by the indirect effect of magnetic fields. The velocity structure is inconsistent with a merger origin or gravitational free-fall of cooling gas in this massive central galaxy. If the molecular clouds originated in gas cooling even a few kpc from their current locations their velocities would exceed those observed. Instead, the projection of the N and SE filaments underneath X-ray cavities suggests they formed in the updraft behind bubbles buoyantly rising through the cluster atmosphere. Direct uplift of the dense gas by the radio bubbles appears to require an implausibly high coupling efficiency. The filaments are coincident with low temperature X-ray gas, bright optical line emission and dust lanes indicating that the molecular gas could have formed from lifted warmer gas that cooled in situ.