A dispenser of precut wiping material, in the form of a reel that can be unwound as a strip, the dispenser comprising: - a housing, - a cover that is movable relative to the housing between an open position and a closed position, and - a module received within the housing; where the module is partially covered by the cover in the closed position, and delimits an outlet passage for the strip of material; the module comprising: - two side walls parallel to each other, between which the reel is positioned, - a guiding device for guiding the strip unwound from the reel, - a flap disposed transversely between the side walls, having two side edges that extend facing the side walls, and a front surface comprising a rainbow-shaped profile consisting of ridges and troughs each extending along an arc on the front surface, and - a plate disposed transversely in front of the flap, movable relative to the side walls between a lowered position and a raised position, and having a rear surface comprising a rainbow-shaped profile consisting of ridges and troughs each extending along an arc on the front surface; the rainbow-shaped profile of the flap and the rainbow-shaped profile of the plate being complementary so that the ridges of the flap are received in the troughs of the plate and the ridges of the plate are received in the troughs of the flap when the plate is in the raised position; the flap and the plate thus delimiting between them a space for guiding and clamping the strip, having a variable configuration, and situated immediately upstream from the outlet passage of the strip; the module comprising a relative clearance system between the flap and the plate, varying the configuration of the space for guiding and clamping the strip, the clearance system comprising: - four transverse rods formed projecting from the side edges of the flap, with two separated rods per side edge, and - four receptacles made in the side walls of the module, with two receptacles per side wall, each receptacle receiving one of the rods of the flap and comprising a longitudinal part and a vertical part, a clearance being thus defined longitudinally and vertically between each rod and the corresponding receptacle. - a housing, - a cover that is movable relative to the housing between an open position and a closed position, and - a module received within the housing; - two side walls parallel to each other, between which the reel is positioned, - a guiding device for guiding the strip unwound from the reel, - a flap disposed transversely between the side walls, having two side edges that extend facing the side walls, and a front surface comprising a rainbow-shaped profile consisting of ridges and troughs each extending along an arc on the front surface, and - a plate disposed transversely in front of the flap, movable relative to the side walls between a lowered position and a raised position, and having a rear surface comprising a rainbow-shaped profile consisting of ridges and troughs each extending along an arc on the front surface; - four transverse rods formed projecting from the side edges of the flap, with two separated rods per side edge, and - four receptacles made in the side walls of the module, with two receptacles per side wall,