  We show existence and uniqueness of regular time-periodic solutions to the
Navier-Stokes problem in the exterior of a rigid body, $\mathscr B$, that moves
by arbitrary (sufficiently smooth) time-periodic translational motion of the
same period, provided the size of the data is suitably restricted. Moreover, we
characterize the spatial asymptotic behavior of such solutions and prove, in
particular, that if $\mathscr B$ has a nonzero net motion identified by a
constant velocity $\bar{\mathbf{\xi}}$ (say), then the solution exhibit a
wake-like behavior in the direction $-\bar{\mathbf{\xi}}$ entirely analogous to
that of a steady-state flow around a body that moves with velocity
$\bar{\mathbf{\xi}}$.
