  Chaotic flows drive mixing and efficient transport in fluids, as well as the
associated beautiful complex patterns familiar to us from our every day life
experience. Generating such flows at small scales where viscosity takes over is
highly challenging from both the theoretical and engineering perspectives. This
can be overcome by introducing a minuscule amount of long flexible polymers,
resulting in a chaotic flow dubbed \textit{elastic turbulence}. At the basis of
the theoretical frameworks for its study lie the assumptions of a spatially
smooth and random-in-time velocity field. Previous measurements of elastic
turbulence have been limited to two-dimensions. Using a novel three-dimensional
particle tracking method, we conduct a microfluidic experiment, allowing us to
explore elastic turbulence from the perspective of particles moving with the
flow. Our findings show that the smoothness assumption breaks already at scales
smaller than a tenth of the system size. Moreover, we provide conclusive
experimental evidence that \textit{ballistic} separation prevails in the
dynamics of pairs of tracers over long times and distances, exhibiting a memory
of the initial separation velocities. The ballistic dispersion is universal,
yet it has been overlooked so far in the context of small scales chaotic flows.
