The existing control-function-based approaches to the identification of firm-level production functions are exclusively concerned with the estimation of single-output production functions despite that, in practice, most firms produce multiple outputs. While one can always opt to employ a single-product specification of the production process by a priori aggregating the firm's outputs, such a formulation is rarely an accurate portrayal of the firm's productive process. This paper extends the control-function-based approach to the structural identification and estimation of firm-level production functions and productivity to the multi-product setting. Specifically, I consider the nonparametric estimation of multi-product production functions. Among other advantages, explicit modeling of multiple outputs allows the identification of cross-output elasticities representing the technological trade-off between individual outputs along the firm's production possibilities frontier, which a traditional single-output production function approach is unable to deliver. To showcase the methodology, I apply it to study the multi-product production technology of Norwegian dairy farms during the 1998-2008 period.