Physical unclonable functions (PUFs) serve as a hardware source of private information that cannot be duplicated and have applications in hardware integrity and information security. Here we demonstrate a photonic PUF based on ultrafast nonlinear optical interactions in a chaotic silicon micro-cavity. The device is probed with a spectrally-encoded ultrashort optical pulse, which nonlinearly interacts with the micro-cavity. This interaction produces a highly complex and unpredictable, yet deterministic, ultrafast response that can serve as a unique "fingerprint" of the cavity and as a source of private information for the device's holder. Experimentally, we extract 17.1-kbit binary keys from six different photonic PUF designs and demonstrate the uniqueness and reproducibility of these keys. Furthermore, we experimentally test exact copies of the six photonic PUFs and demonstrate their unclonability due to unavoidable fabrication variations.