Baby scratching her face?

My ten-week-old daughter has gained control of her hands and is now fascinated by her ability to put her hands to her bottle, suck on her fingers, and unfortunately SCRATCH HER FACE! She scratched her face on two occasions today and drew blood. There wasn't much bleeding, but the scratches were about half an inch each and it broke my heart to see her bleed and cry. I have the mittens and they worked wonderfully when she was fresh out of the womb, but now she can wiggle out of them... even a nicer pair I have that goes up higher on her arm. I am wondering if any other parents found another solution for this problem. I am very worried that she will scratch her eyes or cause more painful sores on her beautiful face. What can I do to prevent scratching? What can I put on her scratches?