Aims. Soft gamma repeaters (SGRs) are highly magnetised neutron stars (magnetars) notable for their gamma-ray and X-ray outbursts. We used near-infrared (NIR) imaging of SGR 0501+4516 in the days, weeks, and years after its 2008 outburst to characterise the multi-wavelength emission, and to obtain a proper motion from our long temporal baseline observations. Methods. We present short- and long-term monitoring of the IR counterpart of SGR 0501+4516 and a measurement of its proper motion. Unlike most magnetars, the source has only moderate foreground extinction with minimal crowding. Our observations began only ∼2 hours after the first activation of SGR 0501+4516 in August 2008 and continued for ∼4 years, including two epochs of Hubble Space Telescope (HST) imaging. The proper motion constraint was improved using a third HST epoch from 10 years later. Results. The NIR and X-rays faded slowly during the first week, which was followed by a steeper power-law decay. The behaviour is satisfactorily fit by a broken power law. Three epochs of HST imaging with a 10-year baseline allowed us to determine the quiescent level and to measure a proper motion of μ = 5.4 ± 0.6 mas yr−1. This corresponds to a low transverse peculiar velocity of v ≃ 51 ± 14 km s−1 (at 2 kpc). The magnitude and direction of the proper motion rules out supernova remnant HB9 as the birth site. We can find no other supernova remnants or groups of massive stars within the region traversed by SGR 0501+4516 during its characteristic lifetime (∼20 kyr). Conclusions. Our observations of SGR 0501+4516 suggest three possibilities: that some magnetars are significantly older than expected, that their progenitors produce low supernova ejecta masses, or that they can be formed through accretion-induced collapse or low-mass neutron star mergers. Although the progenitor of SGR 0501+4516 remains unclear, we propose that SGR 0501+4516 is the best Galactic candidate for a magnetar formed through a mechanism other than massive star core-collapse.