Over recent years, wireless indoor positioning systems (WIPS) have attracted considerable research interest. However, high-performance WIPS proposed in the literature requires that the building have at least three access points (APs). This paper proposes an WIPS using a single fifth-generation (5G) Wi-Fi access point. The proposed method uses beam fingerprints and classification models based on KNN (K-nearest neighbor) and Bayes rule. The beam fingerprint is composed of RSS (Received Signal Strength) samples, collected in some 2D locations of the indoor environment for each beam codebook in the off-line phase. In the online phase, RSS samples of the best beams are collected by user equipment (UE) during the beamtracking process, which are then classified based on beam fingerprints into predefined coordinates. Numerical simulations shown that using the best beam samples, it is possible to locate the stationary user's mobile device with average error less than 2.5 m.