This paper presents the design of a sliding mode controller for networked control systems subject to successive Markovian packet dropouts. This paper adopts the Gilbert-Elliott channel model to describe the temporal correlation among packet losses, and proposes an update scheme to select the assumed available states for use in a sliding mode control law. A technique used in the theory of discrete-time Markov jump linear systems is applied to tackle the effect of the packet losses. This involves introducing a couple of Lyapunov functions dependent on the indicator functions of the instantaneous packet loss, and proving that the sliding mode controller is able to drive the system state trajectories into the neighborhood of the designed integral sliding surface in mean-square sense given that the corresponding Lyapunov inequalities are satisfied. The system is guaranteed thereafter to remain inside the neighborhood of the sliding surface. Simulated case studies are presented to illustrate the effectiveness of the control law.