The use of base-isolation systems represents one of the most effective solutions for the seismic protection of medical facilities, aiming to ensure resilient structural performances as well as to avoid downtime. Nevertheless, traditional base-isolation systems may amplify the vertical seismic response similarly to fixed-base structure inducing significant structural and nonstructural damage, especially in a near-fault area where large vertical spectral accelerations are expected. Aim of the present work is to investigate the effectiveness of a double seismic isolation system for the vertical isolation against near-fault ground motions. An in-series vertical assembly of a high-damping rubber bearing (HDRB) and high-damping rubber layers (HDRLs) is considered, the latter independent of the horizontal and vertical (in tension) responses of the HDRB. A five-storey pavilion of the hospital campus in Avellino, Campania (Italy), is retrofitted with the double base-isolation system in line with the current Italian seismic code. A homemade code is implemented for the nonlinear seismic analysis of the test structure, including a lumped plasticity model of RC frame members and shear, compression and tension nonlinear modelling of the double elastomeric isolation system. In-plane and out-of-plane nonlinear modelling of masonry infills arranged along the interior bays of the perimeter is also considered, while a homemade code is implemented for the wavelet analysis. Nonlinear seismic analysis of the test structure is carried out assuming a set of fifteen near-fault earthquakes extracted from the PEER database, with and without accounting for the vertical seismic component.