Fermenter device, comprising: a fermenter container with a container internal space for receiving a fermentation broth; at least one dynamic cross-flow filtration device comprising at least one filter membrane body set with at least one or more filter membrane bodies, optionally at least one or more filter membrane discs, which each has/have, surrounded by a filter membrane, a body internal space, optionally disc internal space, which is in fluid communication with the container internal space, and which is/are arranged in a movable, optionally rotationally movable, manner in order to generate a cross-flow along the associated filter membrane at the surface of the respective filter membrane in the event of its/their movement, and which are optionally arranged sequentially in a stack manner as well as at a distance from one another in stacking direction; and, associated with the respective filter membrane body set, a respective body connection line, optionally disc connection line, which is fluid-connected to the body internal space of the respective filter membrane body of the associated filter membrane body set; at least one permeate discharge line, which is connected to at least one of the respective body connection line for discharging permeate which has passed through the filter membrane of the respective filter membrane body and via its body internal space into the respective body connection line connected to the at least one permeate discharge line, characterised by a supply gas source, which is fluid-connected to at least one of the respective body connection lines via a supply line for supplying supply gas, which is required by enzymes and/or organisms contained in the fermentation broth, via the respective body connection line connected to the supply gas source and the respective body internal space fluid-connected thereto as well as through the filter membrane surrounding this respective body internal space into the container internal space. a fermenter container with a container internal space for receiving a fermentation broth; at least one dynamic cross-flow filtration device comprising at least one filter membrane body set with at least one or more filter membrane bodies, optionally at least one or more filter membrane discs, which each has/have, surrounded by a filter membrane, a body internal space, optionally disc internal space, which is in fluid communication with the container internal space, and which is/are arranged in a movable, optionally rotationally movable, manner in order to generate a cross-flow along the associated filter membrane at the surface of the respective filter membrane in the event of its/their movement, and which are optionally arranged sequentially in a stack manner as well as at a distance from one another in stacking direction; and, associated with the respective filter membrane body set, a respective body connection line, optionally disc connection line, which is fluid-connected to the body internal space of the respective filter membrane body of the associated filter membrane body set; at least one filter membrane body set with at least one or more filter membrane bodies, optionally at least one or more filter membrane discs, which each has/have, surrounded by a filter membrane, a body internal space, optionally disc internal space, which is in fluid communication with the container internal space, and which is/are arranged in a movable, optionally rotationally movable, manner in order to generate a cross-flow along the associated filter membrane at the surface of the respective filter membrane in the event of its/their movement, and which are optionally arranged sequentially in a stack manner as well as at a distance from one another in stacking direction; and, associated with the respective filter membrane body set, a respective body connection line, optionally disc connection line, which is fluid-connected to the body internal space of the respective filter membrane body of the associated filter membrane body set; at least one permeate discharge line, which is connected to at least one of the respective body connection line for discharging permeate which has passed through the filter membrane of the respective filter membrane body and via its body internal space into the respective body connection line connected to the at least one permeate discharge line, characterised by a supply gas source, which is fluid-connected to at least one of the respective body connection lines via a supply line for supplying supply gas, which is required by enzymes and/or organisms contained in the fermentation broth, via the respective body connection line connected to the supply gas source and the respective body internal space fluid-connected thereto as well as through the filter membrane surrounding this respective body internal space into the container internal space.