There are loads of pin vices available that would do the job e.g. here is a cheap set of 4 ............. Amtech 4 Piece Pin Vice Set - Machine Mart - Machine Mart but that's assuming that your Draper drill has a standard type chuck. If it's the type that takes a hex bit direct into the drive end then you will have to look a bit harder.
Have you googled pin vice?
Yes, there is a very small 3-jaw chuck with a 2.3 mm shank to fit small collet drills, I have one from ebay. If it was for a Draper 12v motor tool, then Dremel ones fit.