X-ray computed tomography of small animals and their organs is an essential tool in basic and preclinical biomedical research. In both phase-contrast and absorption tomography high spatial resolution and short exposure times are of key importance. However, the observable spatial resolutions and achievable exposure times are presently limited by system parameters rather than more fundamental constraints like, e.g., dose. Here we demonstrate laboratory tomography with few-ten μm spatial resolution and few-minute exposure time at an acceptable dose for small-animal imaging, both with absorption contrast and phase contrast. The method relies on a magnifying imaging scheme in combination with a high-power small-spot liquid-metal-jet electron-impact source. The tomographic imaging is demonstrated on intact mouse, phantoms and excised lungs, both healthy and with pulmonary emphysema.