Businesses planning for the post-pandemic world are looking for innovative ways to protect the health and welfare of their employees and customers. Wireless technologies can play a key role in assisting contact tracing to quickly halt a local infection outbreak and prevent further spread. In this work, we present a wearable proximity and exposure notification solution based on a smartwatch that also promotes safe physical distancing in business, hospitality, or recreational facilities. Our proximity-based privacy-preserving contact tracing (P$^3$CT) leverages the Bluetooth Low Energy (BLE) technology for reliable proximity sensing, and an ambient signature protocol for preserving identity. Proximity sensing exploits the received signal strength (RSS) to detect the user's interaction and thus classifying them into low- or high-risk with respect to a patient diagnosed with an infectious disease. More precisely, a user is notified of their exposure based on their interactions, in terms of distance and time, with a patient. Our privacy-preserving protocol uses the ambient signatures to ensure that users' identities be anonymized. We demonstrate the feasibility of our proposed solution through extensive experimentation.