Every so often I drill with bits smaller than 1/16 inch. I only have my handyman-type power drill. It can just about grip the tiny bit.
I grip the drill in the vise (when I had a vise). It's easier to hold the light work steady, than to hold the heavy drill steady.
I cannot always keep the spinning bit on center. I can see it bending visibly. I'm surprised it doesn't break. There is some play in the gearing of the drill. Maybe that looseness permits the bit to wobble so that it survives.
Risk of breakage increases dramatically if the bit is not perpendicular to the work.
Bit drifting also leads to breakage. It can be prevented with one tap with a center punch (or nail), making an indent in the work.
Tiny bits are good candidates for sharpening since there isn't too much metal to remove. I look at the bit under a strong magnifier. I sharpen it on a knife-sharpening stone. I have to pay attention to how I press it down on the stone, in hopes I am grinding the right surfaces.
I once purchased an assortment of small bits mail order. Some sizes are duplicates but that's not so bad for the low price. I can't be sure if some shafts are hard-tempered (more breakable) or soft (more bendable), but bendable shafts seem like a good idea for the tiny sizes.