The very-high energy (VHE, E > 100 GeV) gamma-ray sky shows diverse Galactic and extragalactic source populations. For some sources the astrophysical object class could not be identified so far. The nature (Galactic or extragalactic) of the VHE gamma-ray source HESS J1943+213 is explored. We specifically investigate the proposed near-infrared counterpart 2MASS J19435624+2118233 of HESS J1943+213 and investigate the implications of a physical association. We present K-band imaging from the 3.5 meter CAHA telescope of 2MASS J19435624+2118233. Furthermore, 5 years of Fermi-LAT data were analyzed to search for a high-energy (HE, 100 MeV <E< 100 GeV) counterpart. The CAHA observations revealed that the near-infrared counterpart is extended with an intrinsic half light radius of 2 - 2.5 . These observations also show a smooth, centrally concentrated light profile that is typical of a galaxy, and thus point toward an extragalactic scenario for the VHE gamma-ray source, assuming that the near-infrared source is the counterpart of HESS J1943+213. A high-Sersic index profile provides a better fit than an exponential profile, indicating that the surface brightness profile of 2MASS J19435624+2118233 follows that of a typical, massive elliptical galaxy more closely than that of a disk galaxy. With Fermi-LAT a HE counterpart is found with a power law spectrum above 1 GeV. This gamma-ray spectrum shows a rather sharp break between the HE and VHE regimes. The infrared and HE data strongly favor an extragalactic origin of HESS J1943+213 The source is most likely located at a redshift between 0.03 and 0.45 according to extension and EBL attenuation arguments.
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