Summary: An identifying code of a graph \(\varGamma\) is a subset \(C\) of the vertex set \(V\) such that for each \(x \in V\), the intersection of its closed neighbourhood with \(C\) is nonempty and unique. If \(\varGamma\) is a finite graph, the density of an identifying code \(C\) is defined as \(\frac{| C |}{| V |}\), which naturally extends to a definition of density in certain infinite graphs which are locally finite. Denote by \(d^\ast (\varGamma)\) the infimum of the density of an identifying code of a finite or infinite graph \(\varGamma\). In this paper, we study identifying codes of an infinite Cayley graph \(\varGamma_n\), which is the Cartesian product of an infinite path and a complete graph on \(n\) vertices. We prove that \(d^\ast (\varGamma_3) = \frac{5}{12}\), \(d^\ast (\varGamma_4) = \frac{9}{22}\) and \(d^\ast (\varGamma_n) = \frac{n - 1}{2 n}\) for \(n \geq 5\). As an application, we obtain \(d^\ast (\varGamma)\) if \(\varGamma\) is a connected quintic Cayley graph over a generalized quaternion group.