The Solar System is located within a low-density cavity, known as the Local Bubble, which appears to be filled with an X-ray emitting gas at a temperature of 10   K. Such conditions are too harsh for typical interstellar atoms and molecules to survive. There exists an enigmatic tracer of interstellar gas, known as Diffuse Interstellar Bands (DIB), which often appears as absorption features in stellar spectra. The carriers of these bands remain largely unidentified. Here we report the three-dimensional structure of the Local Bubble using two different DIB tracers (5780 and 5797), which reveals that DIB carriers are present within the Bubble. The map shows low ratios of 5797/5780 inside the Bubble compared to the outside. This finding proves that the carrier of the 5780 DIB can withstand X-ray photo-dissociation and sputtering by fast ions, where the carrier of the 5797 DIB succumbs. This would mean that DIB carriers can be more stable than hitherto thought and that the carrier of the 5780 DIB must be larger than that of the 5797 DIB. Alternatively, small-scale denser (and cooler) structures that shield some of the DIB carriers must be prevalent within the Bubble, implying that such structures may be an intrinsic feature of supernova-driven bubbles.