Short-GRB progenitors could come in various flavors, depending on the nature of the merging compact stellar objects (including a stellar-mass black hole or not) or depending on their ages (millions or billions of years). At a redshift of z = 0.122, the nearly face-on spiral host of the short GRB 080905A is one of the closest short-GRB host galaxies identified so far. This made it a preferred target to explore spatially resolved star formation and to investigate the afterglow position in the context of its star formation structures. We used VLT/MUSE integral-field unit observations, supplemented by ATCA 5.5/9.0 GHz radio-continuum measurements and publicly available HST data, to study the star formation activity in the GRB 080905A host galaxy. The MUSE observations reveal that the entire host is characterized by strong line emission. Using the Hα line flux, we measure for the entire galaxy an SFR of about 1.6 M⊙ yr−1, consistent with its non-detection by ATCA. Several individual star-forming regions are scattered across the host. The most luminous region has a Hα luminosity that is nearly four times as high as the luminosity of the Tarantula nebula in the Large Magellanic Cloud. Even though star-forming activity can be traced as close to about 3 kpc (in projection) distance to the GRB explosion site, stellar population synthesis calculations show that none of the Hα-bright star-forming regions is a likely birthplace of the short-GRB progenitor.