Application of thermoplastic fiber-reinforced lightweight composite materials provides a wide range of advantages that are of particular importance for the mobility sector. UD tapes composed of unidirectionally (UD) oriented inorganic fibers embedded in a thermoplastic matrix represent light-weight materials with high tensile strength. This publication addresses recycling aspects of novel UD tape made of a combination of basalt fibers and different PLA (polylactic acid) formulations. The kinetics of enzyme-based separation of polymer from the fiber were investigated. Different types of UD tapes with a thickness of 270–290 µm reinforced with basalt fiber weight ratios ranging between 51 and 63% were incubated at 37 °C in buffer solution (pH 7.4) containing proteinase K. The influence of enzyme concentration, tape weight per incubation tube, proteinase K activators, and tape types on the rate of enzymatic decomposition was investigated. Enzyme activity was measured by analyzing lactate concentration with lactate dehydrogenase and by measuring weight loss of the composite material. The rate of lactate release increased in the first 30 min of incubation and remained stable for at least 90 min. Weight loss of 4% within 4 h was achieved for a tape with 56% (w/w) fiber content. For a sample with a surface area of 3 cm2 in a buffer volume of 10 mL, the rate of lactate release as a function of enzyme concentration reached saturation at 300 µg enzyme/mL. With this enzyme concentration, the rate of lactate release increased in a linear manner for tape surface areas between 1 and 5 cm2. Four tapes with different PLA types were treated with the enzyme for 17 h. Weight loss ranged between 7 and 24%. Urea at a concentration of 0.5% (w/v) increased lactate release by a factor of 9. Pretreatment of tapes in alkaline medium before enzymatic degradation increased weight loss to 14% compared to 5% without pretreatment. It is concluded that enzymatic PLA hydrolysis from UD tapes is a promising technology for the release of basalt fibers after alkaline pretreatment or for the final cleaning of basalt fibers.