We present new, deep H I line and 20-cm radio-continuum data of the very puzzling blue compact dwarf galaxy NGC 5253, obtained with the Australia Telescope Compact Array as part of the 'Local Volume H I Survey' (LVHIS). Our low-resolution H I maps show, for the first time, the disturbed H I morphology that NGC 5253 possesses, including tails, plumes and detached H I clouds. The high-resolution map reveals an H I plume at the SE and an H I structure at the NW that surrounds an Hα shell. This latter structure is related to an expanding bubble in the interstellar medium (ISM), but it will almost certainly not originate a galactic wind. We confirm that the kinematics of the neutral gas in NGC 5253 are highly perturbed and do not follow a rotation pattern. We discuss the outflow and infall scenarios to explain such disturbed kinematics, analyse the environment in which NGC 5253 resides and compare its properties with those observed in similar star-forming dwarf galaxies. The radio-continuum emission of NGC 5253 is resolved and associated with the intense star-forming region located at the centre of the galaxy. We complete the analysis using multiwavelength data extracted from the literature, which include X-ray, Galaxy Evolution Explorer (GALEX) far-ultraviolet, optical B and R band and Hα, near-infrared H band, and far-infrared data. We estimate the star formation rate using this multiwavelength approach, and compare the results with other galaxy properties. NGC 5253 does not satisfy the Schmidt-Kennicutt law of star formation, has a very low H I mass-to-light ratio when comparing with its stellar mass and seems to be slightly metal-deficient in comparison with starbursts of similar baryonic mass. Taking into account all available multiwavelength data, we conclude that NGC 5253 is probably experiencing the infall of a diffuse, low-metallicity H I cloud along the minor axis of the galaxy. The infall of this independent H I cloud is comprising the ISM and triggering the powerful starburst we see in NGC 5253. The tidally disturbed material observed at the east and north of the galaxy is a consequence of this interaction, which probably started more than 100 Myr ago. The origin of this H I cloud may be related with a strong interaction between NGC 5253 and the late-type spiral galaxy M83 in the past. The observations were obtained with the Australia Telescope which is funded by the Commonwealth of Australia for operations as a National Facility managed by CSIRO.