Do falconers spank or discipline their birds?

No. Falconry or bird-of-prey handling is much more complex than you suggest. Handlers have to have many hours of apprenticeship and training to hunt with the bird. The keyWord is with, they are not really in charge. As for spanking or anything physical, that would be egregious and so dangerous. They can build trust with the bird by giving it bits of food and such, and training much with it, but they absolutely cannot, say, just give a falcon a spank on its rear end when things go awry. That's not at all how it works.