In a recent episode of game of thrones, two sides were engaged in a close battle, one side was substantially larger than the other(3000 vs 6000), while the smaller army was preoccupied with a light infantry force sent by the larger army, the larger army used a large pike force to encircle and slowly advance and kill the force it had surrounded. So, my question is, are there any examples in history where one side used this "pike encirclement" tactic?
No, no one has ever tried to literally encircle an enemy force with pikemen. Outflank them, yes, but the idea of forming a circle of tightly-packed pikemen with an enemy inside is ludicrous. An enemy force would have to be either pathetically tiny or you would have to thin out your own rank of pikemen to an absurd degree to achieve that literal circle of men. A commander would have to spend lots of time in drilling his men to form a big dumb circle, time and energy that would be much better spent on training your pikes to advance in the normal way, which is already effective and complicated enough that you wouldn't want to throw a giant wrench into the works. If the enemy force is small enough and its commander is stupid enough that you could even attempt to do it, you've already won.