ABRIDGED: Diffuse interstellar bands (DIBs) are faint spectral absorption features of unknown origin. Research on DIBs beyond the Local Group (LG) will surely blossom in the era of the ELTs. A possibility that needs to be explored is the use of integral field spectrographs. We do so by using MUSE data for the Antennae Galaxy, the closest major galaxy merger. High S-to-N spectra were created by co-adding the signal of many spatial elements. The emission of the underlying stellar population was modeled using STARLIGHT. To our knowledge, we have derived the first maps for the DIBs at l5780 and l5797 in galaxies outside the LG. The l5780 DIB was detected in an area of ~0.6 arcmin2, corresponding to a linear scale of ~25 kpc2. This region was sampled using >200 independent lines of sight. The DIB l5797 was detected in >100 independent lines of sight. Both DIBs are associated with a region with high emission in the HI 21 cm line, implying a connection between atomic gas and DIBs, as the correlations for the Milky Way also suggest. Conversely, there is mild spatial association between the two DIBs and the molecular gas, in agreement with results for our Galaxy that indicate a lack of correlation between DIBs and molecular gas. The overall structure for the DIB strength distribution and extinction are comparable. Within the system, the l5780 DIB clearly correlates with the extinction. Both DIBs follow the relationship between equivalent width and reddening when data for several galaxies are considered. Unidentified Infrared emission Bands (UIBs, likely caused by PAHs) and the l5780 and l5797 DIBs show similar but not identical spatial distributions. We attribute the differences to extinction effects without necessarily implying a radically different nature of the respective carriers. The results illustrate the enormous potential of integral field spectrographs for extragalactic DIB research.
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