  We are concerned with unbounded sets of $\mathbb{R}^N$ whose boundary has
constant nonlocal (or fractional) mean curvature, which we call CNMC sets. This
is the equation associated to critical points of the fractional perimeter
functional under a volume constraint. We construct CNMC sets which are the
countable union of a certain bounded domain and all its translations through a
periodic integer lattice of dimension $M\leq N$. Our CNMC sets form a $C^2$
branch emanating from the unit ball alone and where the parameter in the branch
is essentially the distance to the closest lattice point. Thus, the new
translated near-balls (or near-spheres) appear from infinity. We find their
exact asymptotic shape as the parameter tends to infinity.
