Free-space imaging with a resolution beyond that dictated by the classical diffraction limit may be achieved with a `Veselago-Pendry' superlens made from a metamaterial possessing a number of specific properties, including a negative refractive index (NRI). Although a planar NRI transmission-line (NRI-TL) metamaterial based on the periodic lumped loading of a host TL network has successfully verified the phenomenon of superlensing in a 2D microstrip environment, a true Veselago-Pendry superlens capable of interacting with and manipulating fields in free space remained elusive, largely due to the difficulty of meeting its stringent design constraints and also to the problem of realizing a full 3D isotropic, polarization-independent structure. This work presents the first experimental verification of free-space Veselago-Pendry superlensing using a new class of volumetric metamaterials based on 2D NRI-TL layers that, although polarization-specific, may be easily constructed using available lithographic techniques to interact with free-space sources. An equivalent-circuit model is developed to enable accurate design of the metamaterial's dispersion and transmission characteristics, including those associated with Veselago-Pendry superlensing, and is validated using full-wave simulations. First, a volumetric NRI-TL metamaterial employing fully printed loading elements is fabricated to verify the salient properties of a free-space metamaterial-slab lens. This lens demonstrates diffraction-limited focusing at X-band and, thus, affirms theoretical results that suggest that electrically thick and lossy metamaterials are unable to perform superlensing. Thereafter, a volumetric NRI-TL metamaterial based on discrete lumped elements is designed to meet the conditions of the Veselago-Pendry superlens at 2.40GHz, and experimentally demonstrates a resolution ability over three times better than that afforded by the classical diffraction limit. A microwave superlens designed in this fashion can be particularly useful for illumination and discrimination of closely spaced buried objects over practical distances by way of back-scattering, for example, in tumour or landmine detection, or for targeted irradiation over electrically small regions in tomography or hyperthermia applications. Possible optical implementations of the volumetric topology are also suggested, and finally, a fully isotropic, polarization-independent 3D metamaterial structure related to the volumetric NRI-TL structure is proposed.