Will I have to chuck my Drill Press Chuck?
That is a Morris taper. Clean it off and slip it back in place a light tap with a mallet should seat it. The tail stock on my lathe has a MT4 much larger and I have spun it out several times, I wipe it down with brake cleaner inside and out and slide it back in then a tap and it is fine. I have a fly cutter that is about 5" and I use it all the time on the mill I use light pressure and don't usually have any problems. Your drill bit probably grabbed and sucked down breaking the connection. Don't drive it in hard or you may deform the mandrel.