Long (>2 s) gamma-ray bursts (GRBs) are associated with explosions of massive stars, although in three instances, supernovae (SNe) have not been detected, despite deep observations. With new H i line and archival optical integral-field spectroscopy data, we characterize the interstellar medium (ISM) of the host galaxy of one of these events, GRB 111005A, in order to shed light on the unclear nature of these peculiar objects. We found that the atomic gas, radio continuum, and rotational patterns are in general very smooth throughout the galaxy, which does not indicate a recent gas inflow or outflow. There is also no gas concentration around the GRB position. The ISM in this galaxy differs from that in hosts of other GRBs and SNe, which may suggest that the progenitor of GRB 111005A was not an explosion of a very massive star (e.g., a compact object merger). However, there are subtle irregularities of the GRB 111005A host (most at a 2 sigma level), which may point to a weak gas inflow or interaction. Because in the SE part of the host there is 15% more atomic gas and half the molecular gas than in the NW part, the molecular gas fraction is low. In the SE part there is also a region with a very high H alpha equivalent width. There is more continuum 1.4 GHz emission to the SE and an S-shaped warp in the UV. Finally, there is also a low-metallicity region 3.'' 5 (1 kpc) from the GRB position. Two galaxies within 300 kpc or a past merger can be responsible for these irregularities.