A classical scenario suggests that ULIRGs transform colliding spiral galaxies into a spheroid dominated early-type galaxy. Recent high-resolution simulations have instead shown that, under some circumstances, rotation disks can be preserved during the merging process or rapidly regrown after coalescence. Our goal is to analyze in detail the ionised gas kinematics in a sample of ULIRGs to infer the incidence of gas rotational dynamics in late-stage interacting galaxies and merger remnants. We analysed MUSE data of a sample of 20 nearby (z<0.165) ULIRGs, as part of the "Physics of ULIRGs with MUSE and ALMA" (PUMA) project. We found that 27% individual nuclei are associated with kpc-scale disk-like gas motions. The rest of the sample displays a plethora of gas kinematics, dominated by winds and merger-induced flows, which make the detection of rotation signatures difficult. On the other hand, the incidence of stellar disk-like motions is ~2 times larger than gaseous disks, as the former are probably less affected by winds and streams. The eight galaxies with a gaseous disk present relatively high intrinsic gas velocity dispersion (sigma = 30-85 km/s), rotationally-supported motions (with gas rotation velocity over velocity dispersion vrot/sigma > 1-8), and dynamical masses in the range (2-7)x1e10 Msun. By combining our results with those of local and high-z disk galaxies from the literature, we found a significant correlation between sigma and the offset from the main sequence (MS), after correcting for their evolutionary trends. Our results confirm the presence of kpc-scale rotating disks in interacting galaxies and merger remnants, with an incidence going from 27% (gas) to ~50% (stars). The ULIRGs gas velocity dispersion is up to a factor of ~4 higher than in local normal MS galaxies, similar to high-z starbursts as presented in the literature.