Are you running your own watchmaking project?
I assume that you used normal High Speed Steel drill, it might become useless soon, because stainless steel is very hard and stiff.
When I make a hole on SUS material, I don't use HSS drill but Vanadium/Cobalt-HSS with titan coated(if possible) one. It is more expensive than the former, but it is very effective and has a long life for SUS work.
And cutting/drilling oil is required. I'm not sure about the thickness you want to go through, you should open the hole little by little, by repeating up and down drill chuck, in higher speed rotation.
Hope this helps. I wish you the best!