The issue for you is that you are impinging a portion of your brachial plexus at the C8 nerve root (ulnar nerve), or that is at least what I am getting from your description.
When you are in the bottom portion of your dip, the brachial plexus is being pressed against the coracoid process of your scapula and the clavicle.
How to fix this? Couple things:
1) Look at your form during dips. If you feel that your shoulders are coming closer together, you really need to concentrate on keeping your chest up more to move the shoulders apart. Especially at the bottom portion of the exercise.
2) Stretch your pecs (clavicular head) and strengthen your lower traps.
That is some really easy stuff I can tell you. However, this is all based on the minimal symptoms you are presenting with. If this doesn't help it, lemme know.