Context. Diffuse interstellar bands (DIBs) are faint spectral absorption features of unknown origin. Research on DIBs beyond the Local Group is very limited and will surely blossom in the era of the Extremely Large Telescopes. However, we can already start paving the way. One possibility that needs to be explored is the use of high-sensitivity integral field spectrographs. Aims. Our goals are twofold. First, we aim to derive reliable mapping of at least one DIB in a galaxy outside the Local Group. Second, we want to explore the relation between DIBs and other properties of the interstellar medium (ISM) in the galaxy. Methods. We use Multi Unit Spectroscopic Explorer (MUSE) data for the Antennae Galaxy, the closest major galaxy merger. High signal-to-noise spectra were created by co-adding the signal of many spatial elements with the Voronoi binning technique. The emission of the underlying stellar population was modelled and substracted with the STARLIGHT spectral synthesis code. Flux and equivalent width of the features of interest were measured by means of fitting to Gaussian functions. Results. To our knowledge, we have derived the first maps for the DIBs at λ5780 and λ5797 in galaxies outside the Local Group. The strongest of the two DIBs (at λ5780) was detected in an area of ~0.6□′, corresponding to a linear scale of ~25 kpc². This region was sampled using >200 out of ~1200 independent lines of sight. The DIB λ5797 was detected in >100 independent lines of sight. Both DIBs are associated with a region of high emission in the H I 21 cm line, implying a connection between atomic gas and DIBs, as the correlations in 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 structures for the DIB strength distribution and extinction are comparable. Within the system, the λ5780 DIB clearly correlates with the extinction, and both DIBs follow the relationship between equivalent width and reddening when data for several galaxies are considered. This relationship is tighter when comparing only with galaxies with metallicities close to solar. Unidentified infrared emission bands (UIBs, likely caused by polycyclic aromatic hydrocarbons PAHs) and the λ5780 and λ5797 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. Conclusions. The results illustrate the enormous potential of integral field spectrographs for extragalactic DIB research.