  The dynamics of a driven tracer in a quiescent bath subject to geometric
confinement effectively models a broad range of phenomena. We explore this
dynamics in a 1D lattice model where geometric confinement is tuned by varying
particle overtaking rates. Previous studies of the model's stationary
properties on a ring of $L$ sites have revealed a phase in which the bath
density profile extends over an $\sim \mathcal{O}(L)$ distance from the tracer
and the tracer's velocity vanishes as $\sim 1/L$. Here, we study the model's
dynamics in this phase as $L\rightarrow \infty$ and for long times. We show
that the bath density profile evolves on a $\sim \sqrt{t}$ time-scale and,
correspondingly, that the tracer's velocity decays as $\sim 1/\sqrt{t}$. Unlike
the well-studied non-driven tracer, whose dynamics becomes diffusive whenever
overtaking is allowed, we here find that driving the tracer preserves its
hallmark sub-diffusive single-file dynamics, even in the presence of
overtaking.
