In this work, we introduce and start the analysis of a periodic and nonlinear system of delay difference equations describing a chemostat with periodic inputs of limiting nutrient and sizestructured biomass. The main novelties of this article are the following: (i) this is the first study of a discrete, structured, and periodic chemostat model taking into account the existence of a time delay between the absorption of nutrient by the biomass cells and its corresponding effects on the cell growth, (ii) we obtain a set of sufficient conditions ensuring the existence of periodic solutions, and (iii) we emphasize that the inclusion of a delay prevents us to follow the standard dimensional reduction and motivated us to carry out an original way to proof the existence of periodic solutions, which is based in a truncation method combined with the use of a theorem by F. Browder on the continuation of fixed points.