Drillman1 has these 0.0310" drill bits that have an ~10mm drilling depth and a 1/8" (3.175mm) shank.
To drill deeper, I have milled a 1/8" channel for the deeper holes when the drill bit I wanted to use could not cut that deep. I think it came out nice, but it depends on the look you are going for as to whether it will work for you.
EDIT: Or, if you are looking for an endmill, and don’t mind going up to a 0.0315" diameter, these bits have a 6.5mm cutting length.
EDIT 2: I have these bits in their “regular length”, “XL” length and “XXL” length, and the only times I break them is running them too fast, or running them through wood with a big difference in hardness between the growth rings. They are fine in bamboo since it is very homogeneous.