A coloring of vertices of a given graph is called perfect if the color structure of each ball of radius $1$ in the graph depends only on the color of the ball center. Let $n$ be a positive integer. We consider a lexicographic product of the infinite path graph and a graph $G$ that can be either the complete or empty graph on $n$ vertices. We give a complete description of perfect colorings with an arbitrary number of colors of such graph products.