So she was 6 weeks old when you got her?
Where did you get her from?
It sounds like she has not been handled much by the breeder and instead has bonded more with the other pups/dogs (hence why she will play so much with your other dog). Its possible she has been reared outside with minimal human contact that would make her react in the way you describe.
You need to be patient. I would try to get her playing with you more than the other dog (give them time out from each other). Spend time just sitting with her on her own, you could do a bit of gentle training using treats, get her to 'sit' etc. then give her the treat and praise her.
Basically you need to build up your relationship with her (away from the other dog), otherwise she will become far too dependent on the other dog and will look to him/her for leadership all the time.
I would avoid picking her up as much as possible, some small dogs hate being picked up.
Try and get her to come to you as much as possible, rather than you approaching her and when she does approach you make it pleasurable for her.
You do need to get it sorted out as in the future she is going to need grooming on a regular basis and possibly clipping as well, so you have to make this as stress free as possible. With this in mind I would start gently grooming her with a soft brush now.