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A spherical capsule (radius$$R$$) is suspended in a viscous liquid (viscosity$$\mu$$) and exposed to a uniaxial extensional flow of strain rate$$E$$. The elasticity of the membrane surrounding the capsule is described by the Skalak constitutive law, expressed in terms of a surface shear modulus$$G$$and an area dilatation modulus$$K$$. Dimensional arguments imply that the slenderness$$\epsilon$$of the deformed capsule depends only upon$$K/G$$and the elastic capillary number$${Ca}=\mu R E/G$$. We address the coupled flow–deformation problem in the limit of strong flow,$${Ca}\gg 1$$, where large deformation allows for the use of approximation methods in the limit$$\epsilon \ll 1$$. The key conceptual challenge, encountered at the very formulation of the problem, is in describing the Lagrangian mapping from the spherical reference state in a manner compatible with hydrodynamic slender-body formulation. Scaling analysis reveals that$$\epsilon$$is proportional to$${Ca}^{-2/3}$$, with the hydrodynamic problem introducing a dependence of the proportionality prefactor upon$$\ln \epsilon$$. Going beyond scaling arguments, we employ asymptotic methods to obtain a reduced formulation, consisting of a differential equation governing a mapping field and an integral equation governing the axial tension distribution. The leading-order deformation is independent of the ratio$$K/G$$; in particular, we find the approximation$$\epsilon ^{2/3} {Ca}\approx 0.2753\ln (2/\epsilon ^2)$$for the relation between$$\epsilon$$and$$Ca$$. A scaling analysis for the neo-Hookean constitutive law reveals the impossibility of a steady slender shape, in agreement with existing numerical simulations. More generally, the present asymptotic paradigm allows us to rigorously discriminate between strain-softening and strain-hardening models.