New to buy collets for Proxxon TBM 115,

This photo is from a Dremel (not Drill Master) but it should be similar. Your tool may have different size collets that take different size diameter shaft bits. You may have the wrong collet (collar) for the bit you want to use.
If your tool only has one size collet then there could be debris stuck in the collet slits which prevent the collet from compressing down on the shaft bit when the top nut is tightened. And sometimes it just takes more torque on the nut that screws over the collet to really tighten it down on the shaft

remove the collet nut
use pliers or the like and pull the collet from the shaft jaws
remove the broken bit
put the collet back in the jaws and refit the nut

This has to be worth a go:

0 3 3 4mm Mini Electric Grinder Keyless Drill Chuck 3 4mm Shank for Dremel...

it is easy, you may find first time that it will not be so, but if you follow advice, it will become even more easy.

1. switch off motor
2. unplug from power
3. find button on the base close to rotating bit, press and hold, this is shaft lock which if applied correctly (you may need to rotate shaft a bit till it fully engage) will stop shaft from rotating
4. apply small spanner (provided with set) to collet collar (collet have two indentations fitting perfectly little spanner)
5. if shaft is held up, turn spanner anticlockwise, then use your hand to undo collar completely
6. remove collet and bit - separate bit from collet
7. collet comes in many sizes, select correct size for new bit
8. insert collet into throat
9. lightly screw in collar
10, insert new bit, at least 2/3 of the bit shank must be inserted into the collet!
11. held shaft lock button pressed
12 hand tighten the collar till bit is locked firmly in place
13. do not use spanner to tighten! if collar do not held properly, re-tighten by hand only!
14. wear always glasses when operating dremmel, or any other rotating machinery, it is not a toy! and exploding bit will penetrate not only your bit of woo, or metal but your eye as well... and your have only two and no spares...

if you find my advice useful, please do not hesitate to vote4 for me, or leave comment, or both... thanx!

it is possible that you stripped lock mechanism on the collet, for the beginning, never, ever, use wrench to tighten the collet! just hand tied, if it slips, tight it bit more, but with hands only.

i never used wrench on the collet and never have any trouble with any of my dremels.

to close, just hold firmly in one hand pressing locket button, tighten up with other hand, to undo, reverse procedure.

if you find bit is too lose, then you have collet inserts, find that one which hold bit firmly and use it on this particular bit. standard collet is 1/8" (3.175mm), but in the set there are collets from 0.0394'' (1mm) to 0.177" (4.5mm). do not use wrench...

just inspect now if thread is not stripped, on spindle and on collar, if locket still hold, if these are damaged, you need to change damaged part. and follow the advice :-)

if you find this advice useful, pleasae do not hesitate leave your remark and vote4 for me, thanx!

You'll need to use either the built in shaft lock to hold the collet in a fixed position, or use a pair of slip joint pliers to hold the shaft of the bit, while you use the wrench to loosen the collet. Slow and steady should get it loose. Another option is use some penetrating oil sprayed lightly onto the back of the collet to help loosen it from the shaft etc.

sounds like the rubber connector between the motor and collet.i use rubber tubing and hot glue to keep it working. bad news is you have to disassemble it to install a new connector.

the collet is either the wrong size for the shaft on the attachment or is missing alltogether

sounds like u are using the wrong collet in the end of dremel the normally have a couple of different collets try changing it for a tighter one also make sure its in the right way

Try using a tapered wood screw or sheet metal screw as an extractor. Choose one of a size that will let its tip go down into the broken collet and get a grip when you twist it. If that fails, you might use a screw extraction tool.