One of the primary challenges in the seismic retrofitting of hospital buildings lies in the research of an effective solution so as not to interrupt 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 and equipped with their own foundation, 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 (F). 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 parallel to all façades of the existing building are selected: i.e. lumped (DEX.L) and distributed (DEX.D), placed along some (with MIs) or all perimetral bays, respectively; mixed (MDEX.L), where DEX.L is combined with a steel EX parallel to the corner perimetral bays. A lumped plasticity model is adopted for RC frame members, while a five-element macro-model is considered for MIs. 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 bare structure, assuming linear and uniform vertical distributions of the seismic load. This is followed by nonlinear dynamic analysis of the infilled structure before and after retrofitting, in which IP and OOP contributions of MIs parallel and perpendicular to the direction of seismic loads, respectively, are considered. The effectiveness and reliability of the proposed DBD procedure of hysteretic DEXs is checked at the at the serviceability and ultimate design earthquakes.