Relativistic plasma jets are observed in many systems that host accreting black holes. According to theory, coiled magnetic fields close to the black hole accelerate and collimate the plasma, leading to a jet being launched1-3. Isolating emission from this acceleration and collimation zone is key to measuring its size and understanding jet formation physics. But this is challenging because emission from the jet base cannot easily be disentangled from other accreting components. Here, we show that rapid optical flux variations from an accreting Galactic black-hole binary are delayed with respect to X-rays radiated from close to the black hole by about 0.1 seconds, and that this delayed signal appears together with a brightening radio jet. The origin of these subsecond optical variations has hitherto been controversial4-8. Not only does our work strongly support a jet origin for the optical variations but it also sets a characteristic elevation of ≲103 Schwarzschild radii for the main inner optical emission zone above the black hole9, constraining both internal shock10 and magnetohydrodynamic11 models. Similarities with blazars12,13 suggest that jet structure and launching physics could potentially be unified under mass-invariant models. Two of the best-studied jetted black-hole binaries show very similar optical lags8,14,15, so this size scale may be a defining feature of such systems.