Our knowledge about the fine structure of lightning processes at Jupiter was substantially limited by the time resolution of previous measurements. Recent observations of the Juno mission revealed electromagnetic signals of Jovian rapid whistlers at a cadence of a few lightning discharges per second, comparable to observations of return strokes at Earth. The duration of these discharges was below a few milliseconds and below one millisecond in the case of Jovian dispersed pulses, which were also discovered by Juno. However, it was still uncertain if Jovian lightning processes have the fine structure of steps corresponding to phenomena known from thunderstorms at Earth. Here we show results collected by the Juno Waves instrument during 5 years of measurements at 125-microsecond resolution. We identify radio pulses with typical time separations of one millisecond, which suggest step-like extensions of lightning channels and indicate that Jovian lightning initiation processes are similar to the initiation of intracloud lightning at Earth.