Cellular solids are remarkably strong structures built from seemingly fragile materials. In order to gain new insight into the mechanical behaviour of these omnipresent materials, we analyse the deformation of seamless cellular bodies within the framework of finite strain elasticity and identify behaviours which are not captured under the small strain regime. Assuming that the cell walls are hyperelastic, we devise a mathematical mechanical strategy based on a successive deformation decomposition by which we approximate the large deformation of periodic cellular structures, as follows: (i) firstly, a uniformly deformed state is assumed, as in a compact solid made from the same elastic material; (ii) then the empty spaces of the individual cells are taken into account by setting the cell walls free. For the elastic structures considered here, an isochoric deformation that can be maintained in both compressible and incompressible materials is considered at the first step, then the stresses in this known configuration are used to analyse the free shape problem at the second step where the cell geometry also plays a role. We find that, when these structures are submitted to uniform external conditions such as stretch, shear, or torsion, internal non-uniform local deformations occur on the scale of the cell dimension. For numerical illustration, we simulate computationally the finite elastic deformation of representative model structures with a small number of cells, which convey the complexity of the geometrical and material assumptions required here. Then the theoretical mechanical analysis, which is not restricted by the cell wall material or number of cells, indicates that analogous finite deformation effects are expected also in other physical or computer models.