Instance of FPT-reductions that is not a polynomial-time reduction
An early example is the W[2]-hardness proof for Tournament Dominating Set (Theorem 4.1 in [1]). The reduction is from Dominating Set and it constructs a tournament with $O(2^k n)$ vertices, where $n$ is the number of vertices of the dominating set instance and $k$ is the parameter.

[1]: Rodney G. Downey and Michael R. Fellows. Parameterized computational feasibility. In
P. Clote and J.B. Remmel, editors, Proceedings of Feasible Mathematics II,
pages 219-244. Birkhauser, 1995.