Kinematic orbit determination offers an efficient and highly accurate alternative to traditional methods by eliminating the need for time-consuming orbit integration and complex satellite dynamics modeling (e.g., solar radiation pressure, earth radiation pressure, etc.). Leveraging the Ka-band inter-satellite link (ISL) payloads deployed on the BeiDou global navigation satellite system (BDS-3), this study presents, for the first time, kinematic orbit determination results for BDS-3 satellites using real ISL measurements. The analysis reveals that the position dilution of precision (PDOP) for Medium Earth Orbit (MEO) satellites ranges from 0.8 to 2.0, while for Inclined Geosynchronous Orbit (IGSO) and Geostationary (GEO) satellites, PDOP values remain within 1.2-2.5 and 1.2-2.0, respectively. The mean 3D RMS values of kinematic orbits are approximately 13.6 cm, 23.5 cm, and 33.7 cm for MEO, IGSO and GEO satellites, respectively, when the orbits of all satellites except one are constrained to the precise dynamic solutions. The mean cross-track accuracy of BDS-3 satellites is 7.3 cm, which is more than 1.5 cm larger than that of the along-track and radial directions. Furthermore, this work systematically investigates the impact of the number of fixed satellites on kinematic solutions, demonstrating that fixing two satellites improves orbit accuracy by 29% over fixing just one, and that constraining all IGSO and GEO satellites yields optimal results for MEO satellites, with mean 3D RMS values of 15.8 cm (along-track), 14.5 cm (crosstrack), and 14.0 cm (radial). Notably, the kinematic orbit accuracy remains robust, as no significant decrease is detected during eclipse seasons.