Chronic wounds represent a significant socio-economic problem, and the improvement of their healing is therefore an essential issue. This paper describes the preparation and biological properties of a novel functionalized nanofiber wound dressing consisting of a polycaprolactone nanofiber carrier modified by a drug delivery system, based on the lipid particles formed by 1-tetradecanol and encapsulated gentamicin and tocopherol acetate. The cytotoxicity of extracts was tested using a metabolic activity assay, and the antibacterial properties of the extracts were tested in vitro on the bacterial strains Staphylococcus aureus and Pseudomonas aeruginosa. The effect of the wound dressing on chronic wound healing was subsequently tested using a mouse model. Fourteen days after surgery, the groups treated by the examined wound cover showed a lower granulation, reepithelization, and inflammation score compared to both the uninfected groups, a lower dermis organization compared to the control, a higher scar thickness compared to the other groups, and a higher thickness of hypodermis and bacteria score compared to both the uninfected groups. This work demonstrates the basic parameters of the safety (biocompatibility) and performance (effect on healing) of the dressing as a medical device and indicates the feasibility of the concept of its preparation in outpatient conditions using a suitable functionalization device.