I have a project where I will take a small tube (say, 11mm in diameter and .5mm thickness) and cut a 9mm band from the end.
Then I will cut a section of that band to make a clip:

I need that clip to be flexible enough to 'clip' onto a rod that is 12mm in diameter (and tapers up to 18mm) yet is firm enough to hold itself in place.
I have been testing carbon fiber, but it just won't budge (literally) when it comes to initially 'clipping' onto the rod. It snaps every time. Other more flexible sources that do clip on end up being stretched and won't hold tight.
Any ideas?

the flexibility highly depends on the wall thickess and the fibre orientation. if you are looking for carbon fibre, a tube with the fibres at +-45° would be the most flexible. it should be no problem to make a laminate, that may clip over the slightly bigger diameter, and still sits quite strong.
however, glass fibres got better absolute elastic  propertys. they are able to flex more without breaking.

Wall thickness needs to be very thin, like .5-.7mm
I need the flexibility to stretch around the rod, but it needs to be firm enough to 'clip' into place and hold tight (not move unless you moved it)
I have never created my own composite tubes before, are you saying I can buy fiberglass fabric sleeves and use my own resin to cure them?
I have never thought of that! I guess if I did that I could control hardness of the resin etc, correct?
If the resin is too hard, it will snap, correct? too soft and it will not hold firm..
Is the same resin used for carbon fiber and fiberglass?

short answer: yes to every question not involving the word "resin".
There are some flexible resins, but this is not, what you are looking for.
does it have to be carbon, or glass fibre? extruded thermoplast tubes might solve your problem much cheaper.
if you want to make them yourself: take some cheap tube of the right external diameter, treat it with some sort of release agent, get a glass fibre sleeve of the right diameter and some standard laminating epoxy and a brush and you are ready to go. mix the epoxy in the right ratio, brush it on you tube, get the sleeve over you tube, and brush out the air bubbles. you are likely looking for 2-3 layers of glass fibres, as those thin sleeves are quite thin. you may want to gently wipe off exess resin, or your tube might get to thick. after curing, take a dremel or saw, and cut out your clips.