Methods are known for producing an anti-resonant hollow-core fiber which has a hollow core extending along a fiber longitudinal axis and an inner jacket region that surrounds the hollow core, said jacket region comprising multiple anti-resonant elements. The known methods have the steps of: providing a cladding tube that has a cladding tube inner bore and a cladding tube longitudinal axis along which a cladding tube wall extends that is delimited by an interior and an exterior; forming a number of precursors for anti-resonant elements at target positions of the cladding tube wall; and elongating the primary preform in order to form the hollow-core fiber or further processing the primary preform in order to form a secondary preform from which the hollow-core fiber is drawn. The aim of the invention is to achieve a high degree of precision and an exact positioning of the anti-resonant elements in a sufficiently stable and reproducible manner on the basis of the aforementioned methods. This is achieved in that the formation of the anti-resonant element precursors includes the formation of elongated pressure chambers, each of which adjoins a wall that can be deformed under pressure and heat in the region of the target positions of the anti-resonant elements and which cause a section of the deformable wall to protrude in the direction of the cladding tube inner bore under the effect of pressure and heat, thereby forming an anti-resonant element or a precursor for same, while carrying out a process according to step (c).