We introduce a Physically Unclonable Function (PUF) based on an ultra-fast chaotic network known as a Hybrid Boolean Network (HBN) implemented on a field programmable gate array. The network, consisting of $N$ coupled asynchronous logic gates displaying dynamics on the sub-nanosecond time scale, acts as a `digital fingerprint' by amplifying small manufacturing variations during a period of transient chaos. In contrast to other PUF designs, we use both $N$-bits per challenge and obtain $N$-bits per response by considering challenges to be initial states of the $N$-node network and responses to be states captured during the subsequent chaotic transient. We find that the presence of chaos amplifies the frozen-in randomness due to manufacturing differences and that the extractable entropy is approximately $50\%$ of the maximum of $N2^{N}$ bits. We obtain PUF uniqueness and reliability metrics $μ_{inter}$ = 0.40$\pm$0.01 and $μ_{intra}$ = 0.05$\pm$0.00, respectively, for an $N=256$ network. These metrics correspond to an expected Hamming distance of 102.4 bits per response. Moreover, a simple cherry-picking scheme that discards noisy bits yields $μ_{intra} < 0.01$ while still retaining $\sim200$ bits/response (corresponding to a Hamming distance of $\sim80$ bits/response). In addition to characterizing the uniqueness and reliability, we demonstrate super-exponential scaling in the entropy up to $N=512$ and demonstrate that PUFmeter, a recent PUF analysis tool, is unable to model our PUF. Finally, we characterize the temperature variation of the HBN-PUF and propose future improvements.