Many industrially important processes feature both nonlinear system dynamics and a process deadtime. Powerful deadtime compensation methods, such as the Smith predictor structure in state space for linear systems is presented first and then directly extended to nonlinear systems. When combined with input/output linearizing state feedback, this Smith-like predictor makes a nonlinear system with deadtime behave like a linear system with deadtime. The control structure is completed by adding an external linear controller, which provides integral action and compensates for the deadtime in the input/output linear system, and an open-loop state observer. Conditions for robust stability with respect to errors in the deadtime and more general linear unstructured multiplicative uncertainties are given. Computer simulations for an example system demonstrate the high controller performance that can be obtained using the proposed method.