We present in situ measurements in a space plasma showing that thin current sheets the size of an ion inertial length exist and are abundant in strong and intermittent plasma turbulence. Many of these current sheets exhibit the microphysical signatures of reconnection. The spatial scale where intermittency occurs corresponds to the observed structures. The reconnecting current sheets represent a type of dissipation mechanism, with observed dissipation rates comparable to or even dominating over collisionless damping rates of waves at ion inertial length scales (x100), and can have far reaching implications for small-scale dissipation in all turbulent plasmas.