Is there gas exchange in the fingers?
At first I was skeptical about this question but then I did some researching and apparently, a certain amount of oxygen is indeed taken up from the surroundings through the upper layers of our skin. In fact, the paper titled "The cutaneous uptake of atmospheric oxygen contributes significantly to the oxygen supply of human dermis and epidermis. states that 


  the upper skin layers to a depth of of 0.25-0.40 mm are almost
  exclusively supplied by external oxygen, whereas the oxygen transport
  of the blood has a minor influence.


Here is another paper that you would find interesting. In short, I would say that its not just the fingers but O2 diffusion into cells take place over the whole body.