One of the primary challenges in the seismic retrofitting of hospital buildings lies in the research of an effective solution so as not to hinder their ability to provide a much-needed service. To this aim, the use of dissipative steel exoskeletons (DEXs), placed in parallel to the façades of the existing structure, represents a viable technique. A displacement-based design (DBD) procedure of DEXs made of concentric braced frames and hysteretic dampers is proposed in the present work, looking at substantially limit structural as well as non-structural damage of the existing framed building. For the latter, attention is focused on the in-plane (IP) and out-of-plane (OOP) nonlinear response of masonry infills (MIs). A five-storey reinforced concrete (RC) pavilion of the hospital campus in Avellino (Italy), with MIs placed in the interior bays of the perimeter frames, is considered as case study and retrofitted with hysteretic DEXs in a high-risk seismic zone. Three external arrangements of DEXs are selected: i.e. lumped (DEX.L) and distributed (DEX.D), placed along some or all perimetral bays, respectively; mixed (MDEX.L), where DEX.L is combined with an EX in the corner perimetral bays. A lumped plasticity model is adopted for RC frame members, based on a piecewise linearisation of the axial load-biaxial bending moment elastic domain of the cross-sections, while a five-element macro-model is considered for MIs, constituted of a central IP truss element and four diagonal OOP beams that govern its inelastic response. Elastic-linear behaviour is assumed for steel frame members of the DEXs, while hysteretic damped braces are modelled with truss elements characterised by a bilinear force-displacement law. The retrofit target displacement is derived from the capacity curve of the original infilled structure, in which IP and OOP contributions of MIs parallel and perpendicular to the direction of seismic loads, respectively, are considered. Nonlinear static analysis is carried out assuming linear and uniform vertical distributions of the seismic load, proportional to the floor masses and the concentrated OOP mass of MIs. This is followed by nonlinear dynamic analysis before and after retrofitting, to assess the effectiveness and reliability of the proposed DBD procedure of hysteretic DEXs. Artificial accelerograms, are considered, matching, on average, the serviceability and ultimate design response spectra provided by the Italian seismic code.