  Thermalization in many-body systems can be inhibited by the application of a
linearly increasing potential, which is known as Stark many-body localization.
Here we investigate the fate of this phenomenon on a two-dimensional
disorder-free lattice with up to $24 \times 6$ sites. Similar to the
one-dimensional case, "density-polarized" regions can act as bottlenecks for
transport and thermalization on laboratory timescales. However, compared to the
one-dimensional case, a substantially stronger potential gradient is needed to
prevent thermalization when an extra spatial dimension is involved. The origin
of this difference and implications for experiments are discussed. We argue
that delocalization is generally favored for typical states in two-dimensional
Stark many-body systems, although nonergodicity can still be observed for a
specific choice of initial states, such as those probed in experiments.
