An array of long, vertically uniform salt fingers in an environment with salt input from above, fresh input from below, a vertically constant, stabilizing temperature gradient and negligible salt diffusion is found to be unstable to perturbations with vertical structure. The maximum growth rate and the form of the instability are derived for fingers with widths that yield maximum buoyancy flux in the unperturbed state. The dependence of the instability on the magnitude of the imposed salt difference is obtained for the heat–salt system. A direct (non-oscillatory) mode with a vertical scale of the order of the buoyancy-layer thickness is the most unstable when the amplitude of the vertical velocity of the fingers is large. The instability is due to the shear flow between rising and sinking fluid in adjacent fingers and is relatively unaffected by the perturbation buoyancy. When the driving is weaker, the dominant instability involves the same processes as for the basic fingers, i.e. perturbation buoyancy, viscosity and diffusion, and the mode becomes oscillatory in time. All of the most unstable modes derived here have a vertical scale of the order of the buoyancy-layer thickness. Both the direct and the oscillatory modes have net horizontal flows that vary with the vertical coordinate and time and in finite amplitude could cause the fingers to incline toward the horizontal. The oscillatory mode involves pairs of fingers so the emerging behaviour could include a kind of period doubling.