The superfluid phases of helium-3 (³He) are predicted to be strongly influenced by mesoscopic confinement. However, mapping out the phase diagram in a confined geometry has been experimentally challenging. We confined a sample of ³He within a nanofluidic cavity of precisely defined geometry, cooled it, and fingerprinted the order parameter using a sensitive nuclear magnetic resonance spectrometer. The measured suppression of the p-wave order parameter arising from surface scattering was consistent with the predictions of quasi-classical theory. Controlled confinement of nanofluidic samples provides a new laboratory for the study of topological superfluids and their surface- and edge-bound excitations.