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5G aims to offer not only significantly higher throughput than previous generations of cellular networks, but also promises millisecond (ms) and sub-millisecond (ultra-)low latency support at the 5G physical (PHY) layer for future applications. While prior measurement studies have confirmed that commercial 5G deployments can achieve up to several Gigabits per second (Gbps) throughput (especially with the mmWave 5G radio), are they able to deliver on the (sub) millisecond latency promise? With this question in mind, we conducted to our knowledge the first in-depth measurement study of commercial 5G mmWave PHY latency using detailed physical channel events and messages. Through carefully designed experiments and data analytics, we dissect various factors that influence 5G PHY latency of both downlink and uplink data transmissions, and explore their impacts on end-to-end delay. We find that while in the best cases, the 5G (mmWave) PHY-layer is capable of delivering ms/sub-ms latency (with a minimum of 0.09 ms for downlink and 0.76 ms for uplink), these happen rarely. A variety of factors such as channel conditions, re-transmissions, physical layer control and scheduling mechanisms, mobility, and application (edge) server placement can all contribute to increased 5G PHY latency (and thus end-to-end (E2E) delay). Our study provides insights to 5G vendors, carriers as well as application developers/content providers on how to better optimize or mitigate these factors for improved 5G latency performance.