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 related to in-plane (IP) and out-of-plane (OOP) collapse of masonry infills (MIs). Nevertheless, traditional base-isolation systems are primarily focused on the horizontal ground motions, but they may amplify the vertical seismic response similarly to fixed-base structure inducing significant structural damage and in turn IP and OOP collapse of MIs at the floor levels. This problem may be worsened in a near-fault area where the vertical excitation generates larger spectral accelerations compared with the horizontal ones, also with a velocity pulse in the vertical direction. Aim of the present work is to investigate the effectiveness of a double seismic isolation system, obtained from the combination of horizontal and vertical base-isolation, 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. An existing five-storey pavilion of the hospital campus in Avellino, Campania (Italy), is retrofitted with the double base-isolation system in line with the provisions of the current Italian seismic code. The RC framed building has a rectangular plan with four and three bays of different length along the longitudinal and transversal directions, respectively, and MIs with two leaves of equal thickness symmetrically arranged in different ways along the exterior and interior bays of the perimeter. A homemade C++ 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. The IP-OOP nonlinear mutual interaction of MIs is represented through a five-element macro-model composed of a central inelastic truss element, with two masses applied at its end sections, and a system of four axially rigid diagonal inelastic beam elements. A homemade MATLAB code is also implemented for the wavelet analysis of vertical seismic floor accelerations of the test structure, to assess the occurrence of vertical moving resonance. Two sets of fifteen near-fault ground motions are extracted from the PEER database, with horizontal pulse-type components rotated in the direction of the strongest pulse and combined with non-pulse type (first set) and pulse-type (second set) vertical component.