In this paper, dead-beat unknown input observers (UIOs) for two-dimensional (2D) state-space models are investigated. Dead-beat UIOs are observers which produce an exact estimate of the original system state trajectory, after a finite number of evolution steps, independently of the system and observer initial conditions and of the unknown disturbances that affect the system functioning. Necessary and sufficient conditions for the existence of dead-beat UIOs are provided, as well as a parameterization of all dead-beat UIO transfer matrices. Comparisons with Luenberger-type UIOs are also carried out, and the extension of the paper results to the case of asymptotic UIOs with a given rate of convergence and to Roesser models are finally discussed.