I have small hands as well, I can totally understand your problem. I also had trouble with doing the fan, but here is some tips, hope it will help.
1.) get bicycle cards. Or as itsurboi99 said plastic cards, you will won't able to do the fan with sticky cards. however take care, if the cards too slippery than it will be a problem, it can fall out of your hands easily.
2.) start with only the half of the deck (or even less). Then you can take more and more cards, but the fan of the whole deck is hard for even professionals with big hand.
3.) you didn't write, but I guess you practiced the Thumb Fan. when I started I wasn't be able to do that. Then for a trick, I needed to learn the Under Pressure flourish (Calen Morelli's Clench), and it required knowing the pressure fan. When I tried, woooow, my fan was much better than the fan I created with the thumb fan. My point is you have to look for several fan-technics, than you can decide which fits you the best. (Since then after getting my pressure fan to a level, I practiced the Thumb fan, and I realized it's much easier)
4.) Practice and don't give up. Once you will feel, how to do the fan, and after that it's gonna be much easier. In addition, who cares about the fan? It's just a fancy flourish, however if you show an awesome trick, nobody will care about if you are able to do the fan.
Hope it helped, have fun with the practicing!