The binary brown dwarf WISE J104915.57$-$531906.1 (also Luhman 16AB), composed of a late L and early T dwarf, is a prototypical L/T transition flux reversal binary located at only 2 pc distance. Luhman 16B is a known variable whose light curves evolve rapidly. We present spatially resolved spectroscopic time-series of Luhman 16A and B covering 6.5 h using HST/WFC3 at 1.1 to 1.66 $mu$m. The small, count-dependent variability of Luhman 16A at the beginning of the observations likely stems from instrumental systematics; Luhman 16A appears non-variable above $approx$0.4%. Its spectrum is well fit by a single cloud layer with intermediate cloud thickness (f_sed=2, Teff=1200 K). Luhman 16B varies at all wavelengths with peak-to-valley amplitudes of 7-11%. The amplitude and light curve shape changes over only one rotation period. The lowest relative amplitude is found in the deep water absorption band at 1.4 $mu$m, otherwise it mostly decreases gradually from the blue to the red edge of the spectrum. This is very similar to the other two known highly variable early T dwarfs. A two-component cloud model accounts for most of the variability, although small deviations are seen in the water absorption band. We fit the mean spectrum and relative amplitudes with a linear combination of two models of a warm, thinner cloud (Teff=1300 K, fsed=3) and a cooler, thicker cloud (Teff=1000-1100 K, f_sed=1), assuming out-of-equilibrium atmospheric chemistry. A cloud as for Luhman 16A but with holes cannot reproduce the variability of Luhman 16B, indicating more complex cloud evolution through the L/T transition. The projected separation of the binary has decreased by $approx$0.3 in 8 months.
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