This paper studies multiobjective optimal control problems in the continuous-time framework when the space of states and the space of controls are infinite-dimensional and with lighter smoothness assumptions than the usual ones. The paper generalizes to the multiobjective case existing results for single-objective optimal control problems in that framework. The dynamics are governed by differential equations and a finite number of terminal equality and inequality constraints are present. Necessary conditions of Pareto optimality are provided namely Pontryagin maximum principles in the strong form. Sufficient conditions are also provided.