  We study properties of quasihyperbolic geodesics on Banach spaces. For
example, we show that in a strictly convex Banach space with the Radon-Nikodym
property, the quasihyperbolic geodesics are unique. We also give an example of
a convex domain $\Omega$ in a Banach space such that there is no geodesic
between any given pair of points $x, y \in \Omega\,.$ In addition, we prove
that if $\mathrm{X}$ is a uniformly convex Banach space and its modulus of
convexity is of a power type, then every geodesic of the quasihyperbolic
metric, defined on a proper subdomain of $\mathrm{X}$, is smooth.
