It's possible you might have a pre-existing problem that you might not have known you had, so after 5 days of this, you should see your doctor about it.

That being said, this is not a normal thing to have. Are you riding a bike that has a flat bar (the entire history of cycle touring has proven this is not a good idea!)? Are you moving your hands around on purpose (which you should do every 10 minutes or so)?

Are you properly-balanced fore-and-aft on your bike? I ask, because it sure sounds like you could be putting too much weight on your hands.