Leading neuroimaging studies have pushed 3T MRI acquisition resolutions below 1.0 mm for improved structure definition and morphometry. Yet, only few, time-intensive automated image analysis pipelines have been validated for high-resolution (HiRes) settings. Efficient deep learning approaches, on the other hand, rarely support more than one fixed resolution (usually 1.0 mm). Furthermore, the lack of a standard submillimeter resolution as well as limited availability of diverse HiRes data with sufficient coverage of scanner, age, diseases, or genetic variance poses additional, unsolved challenges for training HiRes networks. Incorporating resolution-independence into deep learning-based segmentation, i.e., the ability to segment images at their native resolution across a range of different voxel sizes, promises to overcome these challenges, yet no such approach currently exists. We now fill this gap by introducing a Voxelsize Independent Neural Network (VINN) for resolution-independent segmentation tasks and present FastSurferVINN, which (i) establishes and implements resolution-independence for deep learning as the first method simultaneously supporting 0.7-1.0 mm whole brain segmentation, (ii) significantly outperforms state-of-the-art methods across resolutions, and (iii) mitigates the data imbalance problem present in HiRes datasets. Overall, internal resolution-independence mutually benefits both HiRes and 1.0 mm MRI segmentation. With our rigorously validated FastSurferVINN we distribute a rapid tool for morphometric neuroimage analysis. The VINN architecture, furthermore, represents an efficient resolution-independent segmentation method for wider application