I always drill a short distance, then pull the entire tailstock out and let the bit cool.
That is, I loosen the lock handle on the tailstock and pull it completely back while the lathe is turning. If you stop the lathe, the bit can get stuck, and certainly it will be much more difficult to back out.
After the bit is out, I backup the tailstock screw so that it is ready for the next small push. When the bit is cool, I turn on the lathe, then push the tailstock until the bit bottoms out in the back of the hole. I quickly lock the tailstock lever, crank the tailstock to drill another 1/2 inch or so, then repeat.
I do not run the lathe at high speed, but not at the slowest speed either. I use the Fisch pen blank drills for the most part. They are fancy brad point bits. I am undecided whether they are better than ordinary brad point bits or other bits for drilling pen blanks. I think not.
I have not needed a lubricant or coolant, and the process has worked well for me on standard "ribbon" type acrylics, the inlace acrylester, and similar plastic blanks.