Periodic unit-cell solutions are obtained for the finite-strain, elasto-plastic response of a filled closed-cell Kelvin foam in uniaxial compression. The closed-cell Kelvin foam has edges of equal length, and attention is focussed on the regular Kelvin foam with faces comprising regular hexagons and squares. The elongated Kelvin foam is also studied: its faces comprise elongated hexagons and quadrilaterals. Both the cell walls and core of the Kelvin foam are treated as elastic, ideally plastic von Mises solids. In the first part of the study, the core modulus and strength are sufficiently small for the core to behave as an inviscid, incompressible fluid. Filling of the closed-cell Kelvin foam, in regular or elongated form, with an inviscid, incompressible core elevates its yield strength slightly and stabilises the post-yield response against softening. In the second part of the study, the macroscopic modulus and strength of a filled closed-cell foam are determined as a function of core modulus and deviatoric strength. The deformation mode of the cell edges switches from a bending mode to an affine stretching mode when the core is sufficiently stiff and strong; an analytical model is derived for affine deformation of cell walls and core. Finally, the response of a finite specimen containing an edge imperfection is in good agreement with the periodic, unit cell response of the filled Kelvin foam.