A few years ago I wanted to drill small (0.032”) holes into stainless tube. This was before I had machine tools, but I’m not sure my approach would be different. The bit was so small that my drill press was not useful (too slow and not enough feel). I used a Dremel with its little 3-jaw chuck. My tube was thick enough (0.035” wall) that I made some transverse slices to thin where I drilled, as shown below. The bit noticeably degraded but made it through the six holes I wanted.
I’m not sure how helpful it is to your example, but I’d want a pretty fast spindle (5-8k for the 0.032” drill I used) and something with feel. I know there are “sensitive drill presses” that seem well suited to it. My current drill press will go to 4K rpm, but I don’t think there’s enough feel to avoid breaking a bit.
This is an “atomizer” that goes across the nozzle of a backpack leaf blower and lets me dispense insecticide as a mist; for mosquitoes.