  In this paper we study the regular prism tilings and construct ball packings
by geodesic balls related to the above tilings in the projective model of
$\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism
groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism
tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine
packing density formulae for geodesic ball packings generated by the above
prism groups. Moreover, studying these formulae we determine the conjectured
maximal dense packing arrangements and their densities and visualize them in
the projective model of $\mathbf{Nil}$ geometry. We get a dense (conjectured
locally densest) geodesic ball arrangement related to the parameters
$(p,q)=(6,3)$ where the kissing number of the packing is $14$, similarly to the
densest lattice-like $\mathbf{Nil}$ geodesic ball arrangement investigated by
the second author .
