It is not a real good idea to drill a hole in the arbor, and it might be quite difficult to actually do with the round, fairly hard surface and the chuck installed on it. Open the jaws of the chuck and look down inside with a flashlight. Can you see the end of the arbor down in the hole? If so, you can press or drift it out. If you can't see the arbor down in the chuck, then it matters what kind of chuck it is. Take a look, get the chuck brand and model, and maybe we can be more help to you.