The paper considers an incompressible and irrotational flow of a fluid with constant density bounded below by a rigid horizontal bottom and above by a free surface under the influence of gravity and surface tension. It is known that the full Euler equations have travelling two-dimensional solitary-wave solutions of small amplitude for large surface tension (Bond number greater than ⅓). This paper shows that these waves are linearly unstable to three-dimensional perturbations which oscillate along the wave crest with wavenumber in a finite band. The growth rates of these unstable modes are well approximated using the linearized Kadomtsev–Petviashvili equation with positive disper