With millimeter wave (mmWave) wireless communication envisioned to be the key enabler of next generation high data rate wireless networks, security is of paramount importance. While conventional security measures in wireless networks operate at a higher layer of the protocol stack, physical layer security utilizes unique device dependent hardware features to identify and authenticate legitimate devices. In this work, we identify that the manufacturing tolerances in the antenna arrays used in mmWave devices contribute to a beam pattern that is unique to each device, and to that end we propose a novel device fingerprinting scheme based on the unique beam patterns used by the mmWave devices. Specifically, we propose a fingerprinting scheme with multiple access points (APs) to take advantage of the rich spatial-temporal information of the beam pattern. We perform comprehensive experiments with commercial off-the-shelf mmWave devices to validate the reliability performance of our proposed method under various scenarios. We also compare our beam pattern feature with a conventional physical layer feature namely power spectral density feature (PSD). To that end, we implement PSD feature based fingerprinting for mmWave devices. We show that the proposed multiple APs scheme is able to achieve over 99% identification accuracy for stationary LOS and NLOS scenarios and significantly outperform the PSD based method. For mobility scenarios, the overall identification accuracy is 96%. In addition, we perform security analysis of our proposed beam pattern fingerprinting system and PSD fingerprinting system by studying the feasibility of performing impersonation attacks. We design and implement an impersonation attack mechanism for mmWave wireless networks using state-of-the-art 60 GHz software defined radios. We discuss our findings and their implications on the security of the mmWave wireless networks.