We study the total and baryonic mass distributions of the deflector SDSS J0100+1818 through a full strong lensing analysis. The system is composed of an ultra-massive early-type galaxy at z=0.581, with a total stellar mass of (1.5 +/- 0.3)x10(12)M(circle dot) and a stellar velocity dispersion of (450 +/- 40) km s(-1), surrounded by ten multiple images of three background sources, two of which are spectroscopically confirmed at z=1.880. We took advantage of high-resolution HST photometry and VLT/X-shooter spectroscopy to measure the positions of the multiple images and performed a strong lensing study with the software GLEE. We tested different total mass profiles for the lens and modeled the background sources first as point-like and then as extended objects. We successfully predict the positions of the observed multiple images and reconstruct over approximately 7200 HST pixels the complex surface brightness distributions of the sources. We measured the cumulative total mass profile of the lens and find a total mass value of (9.1 +/- 0.1)x10(12)M(circle dot), within the Einstein radius of approximately 42 kpc, and stellar-over-total mass fractions ranging from (49 +/- 12)%, at the half-light radius (R-e=9.3 kpc) of the lens galaxy, to (10 +/- 2)%, in the outer regions (R=70 kpc). These results suggest that the baryonic mass component of SDSS J0100+1818 is very concentrated in its core and that the lens early-type galaxy (or group) is immersed in a massive dark matter halo, which allows it to act as a powerful gravitational lens, creating multiple images with exceptional angular separations. This is consistent with what has been found in other ultra-high-mass candidates at intermediate redshift. We also measured the physical sizes of the distant sources, resolving them down to a few hundred parsecs. Finally, we quantify and discuss a relevant source of systematic uncertainties on the reconstructed sizes of background galaxies, associated with the adopted lens total mass model.