Aims. We investigate the influence of the interstellar medium (ISM) on debris discs using a statistical approach. We probe the effect of the ISM on debris disc occurrence rates and on the morphologies of the discs. Methods. We used results from the Herschel Space Observatory DUNES and DEBRIS surveys of 295 nearby FGK dwarf stars imaged at 100 µm and 160 µm. Most of the 48 debris discs in this sample have small optical depths, making them more likely to be affected by the ISM compared to optically thick discs. Since the stars in our sample are located within the Local Interstellar Cloud, we can infer that their debris discs encounter similar conditions. This allows us to use the stellar space velocity, in particular the U component, as a single indicator of the forces that can act on the debris disc dust grains when they interact with the ISM. Because older stars show a larger dispersion of space velocity values, we investigated the impact of the debris disc ages on our results. Results. The observed debris disc occurrence rates seem to depend on the stellar space velocities, as expected under the hypothesis that stars with higher space velocities have a higher probability of losing their circumstellar dust. The percentage of sources with debris discs in our sample reaches a maximum of ≈25% for stars with low space velocity component values, |Urel|, relative to the local ISM, and decreases monotonically for larger |Urel| values down to the 10% level. A decrease in the average disc fractional luminosity as a function of |Urel| is also observed. These dependences do not disappear after accounting for the reported higher dispersion of U values with age. In extended discs, the impact of the ISM could also explain the links observed between the stellar space velocities and the debris disc's projected ellipticities, position angles, and radii. The fractional luminosities of the debris discs appear to be correlated with their position angles, suggesting that the effect of the ISM on the dust content depends on the disc orientation. Although these indications may not be fully conclusive on their own, they collectively reinforce the hypothesis that the ISM influences the occurrence rates and morphologies of debris discs, thereby motivating additional research on the impact of the environment.