The fine structure and dynamics of sunspots and the strong outflow in their outer filamentary part--the penumbra--have puzzled astronomers for more than a century. Recent theoretical models and three-dimensional numerical simulations explain the penumbral filaments and their radiative energy output as the result of overturning convection. Here, we describe the detection of ubiquitous, relatively dark downward flows of up to 1 kilometer per second (km/s) in the interior penumbra, using imaging spectropolarimetric data from the Swedish 1-meter Solar Telescope. The dark downflows are omnipresent in the interior penumbra, distinguishing them from flows in arched flux tubes, and are associated with strong (3 to 3.5 km/s) radial outflows. They are thus part of a penumbral convective flow pattern, with the Evershed flow representing the horizontal component of that convection.