1 IntroductionIn this paper we consider only simple undirected graphs. We use for terminology andnotation not defined here. Let distG(x, y) denote the distance between vertices x andy in G. The Cartesian product GH of graphs G and H is the graph with vertex setV (G) × V (H), where vertices (x, y) and (x′, y′) are adjacent whenever xx′∈ E(G) andy = y′, or x = x′and yy′∈ E(H). For two infinite paths P1, P2, the Cartesian productP1P2 is usually called the (infinite) square lattice.The concept of the packing colouring comes from the area of frequency planning inwireless networks, and was introduced by Goddard et al. in under the name broadcastcolouring. The packing chromatic number of a graph G, denoted by χp(G), is the smallestinteger k such that V (G) can be partitioned into k disjoint sets X1, . , Xk, where foreach pair of vertices x, y ∈ Xi distG(x, y) > i, for every i = 1,...