The invention relates to a system (100) for delivering frozen or chilled beverages. The system (100) comprises a device (10) and a container (20), the container (20) comprising a frozen product and the device (10) being configured for receiving the container (20) and for processing the frozen product of the container (20) to deliver the final beverage. The device (10) comprises: a slicing element (60), the slicing element (60) and the frozen product being relatively rotatable to each other about a rotation axis in order to slice the frozen product, and a driving unit (50) configured to provide the relative rotation between the slicing element (60) and the frozen product. The container (20) comprises a container inner wall (301) being in contact with the frozen product, wherein the container inner wall (301) has a cross-section, which is shaped such that a relative rotation between the container (20) and the frozen product about the rotation axis is blocked.