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

Geodesic ball packings generated by regular prism tilings in $\mathbf{Nil}$ geometry