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Fully-localised solitary waves propagating on the surface of a three-dimensional ideal fluid of arbitrary depth, and bounded above by an elastic sheet that resists flexing, are computed. The cases of shallow and deep water are distinct. In shallow water, weakly nonlinear modulational analysis (see Milewski & Wang) predicts waves of arbitrarily small amplitude and these are found numerically. In deep water, the same analysis rules out the existence of solitary waves bifurcating from linear waves, but, nevertheless, we find them at finite amplitude. This is accomplished using a continuation method following the branch from the shallow regime. All solutions are computed via a fifth-order Hamiltonian truncation of the full ideal free-boundary fluid equations. We show that this truncation is quantitatively accurate by comparisons with full potential flow in two-dimensions.