The SPT0311-58 system at z=6.900 is an extremely massive structure within the reionization epoch, and offers a chance to understand the formation of galaxies in an extreme peak in the primordial density field. We present 70mas Atacama Large Millimeter/submillimeter Array observations of the dust continuum and CII 158um emission in the central pair of galaxies and reach physical resolution ~100-350pc, among the most detailed views of any reionization-era system to date. The observations resolve the source into at least a dozen kiloparsec-size clumps. The global kinematics and high turbulent velocity dispersion within the galaxies present a striking contrast to recent claims of dynamically cold thin-disk kinematics in some dusty galaxies just 800Myr later at z~4. We speculate that both gravitational interactions and fragmentation from massive parent disks have likely played a role in the overall dynamics and formation of clumps in the system. Each clump individually is comparable in mass to other 6Comment: Accepted for publication in ApJ Letters, 9 pages, 5 figures. Interactive versions of Fig. 2 and data products are available at https://github.com/spt-smg/publicdata