As part of our study on the impact of violent star formation on the interstellar medium (ISM) of dwarf galaxies, we report observations of neutral atomic hydrogen (HI) in the post-starburst dwarf galaxy NGC 1569. High-resolution measurements with the VLA (B-, C- and D-array) are aimed at identifying morphological and kinematical signatures in the HI caused by the starburst. Our kinematical data suggest a huge hole in the HI distribution, probably due to the large number of supernovae explosions in the center of the galaxy over the past 20 Myr. Investigating the large-scale HI structure, we confirm the existence of a possible HI companion and a so-called HI bridge east of NGC 1569. Furthermore, we report the detection of additional low-intensity HI halo emission, which leads us to suggest a revised halo structure. Based on the new picture, we discuss the origin of the halo gas and possible implications for the evolution of the starburst in NGC 1569.