The design of an extended Luenberger observer is considered to solve the state observation problem for semilinear distributed-parameter systems. For this, a backstepping-based technique is proposed for the design of the output injection weights by making use of the (extended) linearization of the semilinear observer error system with respect to the observer state. Stability of both the linearized and the semilinear observer error dynamics is analyzed theoretically. Moreover, an efficient sample-and-hold implementation is considered to improve the computational efficiency of the observer design. Simulation examples are provided for a bistable semilinear partial differential equation and the simplified model of a bioreactor with Monod kinetics.