It is hard to find examples of 3-player Nash games along with their Nash equilibrium results. Today, Satish pointed me to the Boston College website.
The Boston College publishes a set of exercises on computing the Nash equilibrium in 2-player and 3-player games, along with the results expected. So, I finally got hold of a few more test cases to double-check my library implementation. Unfortunately, the exercises contain only two 3-player games.