How to further disassemble cordless drill/driver chuck?
Presuming that this type of chuck, sometimes known as a Jacobs chuck, is assembled in a manner similar to that found on drill presses, one can expect that the outer retaining shell is a press fit to the body. 

The trick on such a small chuck is supporting the outer shell while applying extraction force to the jaws.

Many videos abound on the 'net using the terms "restore jacobs chuck" and all of them should cover the extraction process. I found one creator who used a hammer and a vise:

https://youtu.be/hCgWNpE-STk?t=111

but with the thin shell on your hand drill chuck, that doesn't seem quite stable enough. One video creator had a piece of scrap(?) steel tubing with just the right inside diameter to allow him to use a ram type press to gently force the mechanism free.

View many of the videos you'll find with the above search and you will have a more solid approach to your goal.