Context. We map the interstellar medium (ISM) including the diffuse interstellar bands (DIBs) in absorption toward the globular cluster NGC 6397 using VLT/MUSE. Assuming the absorbers are located at the rim of the Local Bubble we trace structures on the order of mpc (milliparsec, a few thousand AU). Aims: We aimed to demonstrate the feasibility to map variations of DIBs on small scales with MUSE. The sightlines defined by binned stellar spectra are separated by only a few arcseconds and we probe the absorption within a physically connected region. Methods: This analysis utilized the fitting residuals of individual stellar spectra of NGC 6397 member stars and analyzed lines from neutral species and several DIBs in Voronoi-binned composite spectra with high signal-to-noise ratio (S/N). Results: This pilot study demonstrates the power of MUSE for mapping the local ISM on very small scales which provides a new window for ISM observations. We detect small scale variations in Na I and K I as well as in several DIBs within few arcseconds, or mpc with regard to the Local Bubble. We verify the suitability of the MUSE 3D spectrograph for such measurements and gain new insights by probing a single physical absorber with multiple sight lines.