The functioning of the Diary Supply Chain (DSC) can be disrupted by many human, environmental, and technical factors. The aim of the work is to develop a model that enables risk analysis of the entire DSC using a fuzzy fault tree. A fuzzy fault tree (FFT) was selected to perform the risk analysis in order to overcome problems related to the unavailability of data regarding failure rates of basic events and problems related to the reluctance of experts to provide exact values of the probability of occurrence of individual basic events. Four FFTs reflecting the most common DSC links were developed to capture problems with the production of basic products, feed and feed components, problems in milk producers and milk processing plants, transport and distribution problems, and problems with wholesale and retail. Fifty-one unwanted basic events were identified. The proposed model is able to carry out fuzzification process by creating membership functions for a group of experts, develop Fuzzy Fault Tree structures, calculate fuzzy risks for unwanted events, generate distribution of the fuzzy risk connected with the top event occurrence, carry out defuzzification, and calculate crisp risk of top event occurrence. The practical application of the proposed risk assessment model was shown in an example DSC. The proposed model gives a helping hand to the managers and employees involved in DSC, helping them meet the challenges of operating DSC demands related to food safety, quality, deadlines, costs and legal issues.