  We present a loophole-free violation of local realism using entangled photon
pairs. We ensure that all relevant events in our Bell test are spacelike
separated by placing the parties far enough apart and by using fast random
number generators and high-speed polarization measurements. A high-quality
polarization-entangled source of photons, combined with high-efficiency,
low-noise, single-photon detectors, allows us to make measurements without
requiring any fair-sampling assumptions. Using a hypothesis test, we compute
p-values as small as $5.9\times 10^{-9}$ for our Bell violation while
maintaining the spacelike separation of our events. We estimate the degree to
which a local realistic system could predict our measurement choices.
Accounting for this predictability, our smallest adjusted p-value is $2.3
\times 10^{-7}$. We therefore reject the hypothesis that local realism governs
our experiment.
