The present study focuses on the elimination of delamination during the drilling of a linen-based biocomposite material in epoxy resin used for the manufacture of sports kayaks, depending on the tool material, cutting conditions, and the use of additional wooden support plates. In the present study, HSS (high-speed steel) and Carbide cutting tools without coatings, with the same geometry and two types of cutting conditions (n = 1500 rpm, fn = 0.05 and 0.1 mm/rev) were used. A Sololite-type wooden backing plate was used to aid in reducing delamination. The results show that the additional support plates significantly reduced delamination by up to 80% both at the material inlet and especially at the drill hole outlet. In this study, the use of a lower feed rate (fn = 0.05 mm/rev) per tooth was shown to have a significant effect on reducing the delamination of biocomposite materials with flax fibers, which are generally known to be difficult to machine. The Carbide cutting tool shows significantly better results both in terms of its wear and in terms of delamination of the biocomposite material. The highest delamination was obtained without the use of a backing board at the tool exit after 50 drilled holes of 3509 µm. With the use of a backing board, this delamination decreased to 693 µm after 50 drilled holes.