  The dynamics of binary stars provides a unique avenue to gather insight into
the study of the structure and dynamics of star clusters and galaxies. In this
paper, we present the results of a set of $N$-body simulations aimed at
exploring the evolution of binary stars during the early evolutionary phases of
ultra-faint dwarf galaxies (UFD). In our simulations, we assume that the
stellar component of the UFD is initially dynamically cold and evolves towards
its final equilibrium after undergoing the violent relaxation phase. We show
that the early evolutionary phases of the UFD significantly enhance the
disruption of wide binaries and leave their dynamical fingerprints on the
semi-major axis distribution of the surviving binaries as compared to models
initially in equilibrium. An initially thermal eccentricity distribution is
preserved except for the widest binaries for which it evolves towards a
superthermal distribution; for a binary population with an initially uniform
eccentricity distribution, memory of this initial distribution is rapidly lost
for most binaries as wider binaries evolve to approach a thermal/superthermal
distribution. The evolution of binaries is driven both by tidal effects due to
the potential of the UFD dark matter halo and collisional effects associated to
binary-binary/single star encounters. Collisional effects are particularly
important within the clumpy substructure characterizing the system during its
early evolution; in addition to enhancing binary ionization and evolution of
the binary orbital parameters, encounters may lead to exchanges of either of
the primordial binary components with one of the interacting stars.
