The long and relatively faint gamma-ray burst GRB 060605 detected by Swift/BAT lasted about 20 s. Its afterglow could be observed with Swift/XRT for nearly 1 day, while Swift/UVOT could detect the afterglow during the first 6 h after the event. Here, we report on integral field spectroscopy of its afterglow performed with PMAS/PPak mounted at the Calar Alto 3.5 m telescope. In addition, we report on a detailed analysis of XRT and UVOT data and on the results of deep late-time VLT observations that reveal the GRB host galaxy. We find that the burst occurred at a redshift of z = 3.773, possibly associated with a faint, RC = 26.4 ± 0.3 host. Based on the optical and X-ray data, we deduce information on the SED of the afterglow, the position of the cooling frequency in the SED, the nature of the circumburst environment, its collimation factor, and its energetics. We find that the GRB fireball was expanding into a constant-density medium and that the explosion was collimated with a narrow half-opening angle of about 2.4 degrees. The initial Lorentz factor of the fireball was about 250; however, its beaming-corrected energy release in the gamma-ray band was comparably low. The optical, X-ray afterglow, on the other hand, was rather luminous. Finally, we find that the data are consistent within the error bars with an achromatic evolution of the afterglow during the suspected jet break time at about 0.27 days after the burst. Based on observations collected at the German-Spanish Calar Alto Observatory in Spain (Programme F06-3.5-055) and at the European Southern Observatory, La Silla and Paranal, Chile (ESO Programme 177.D-0591).