I mostly play with a pick. When I was briefly playing with a metal band, I had some right hand grip pain while doing a video shoot - must've played that track 20 times the first day of shooting, for all the cam angles and special effect/set shots. The next day - OOOWWWCCHHHH - I couldn't lift up a quart of milk.
Yes, the trouble was caused by over grip and repetitive stress of so much 16th note double picking. We mimed to the studio track, so I played too hard trying to keep in sync, hear myself a little, as well as keep the visusal energy up from take to take.
Naturally, doing that style, imho, is mostly pick playing, for me fingers won't do it for the edge, aggression and attack.
My fix was that the producer's assistant was a trigger point therapist, she fixed me in about 5 minutes! It was miraculous, after a whole morning of pain before the second day of shooting. The whole 2nd session was pain free.
I agree with those above that it's too much pressure while picking. In metal, it's easy to get carried away. Another factor is, if you can't CLEARLY hear your bass - not so much volume, but clarity of your picking, you're bound to play harder and grip more intensely as you play, especially on the more emotionally intense musical bits.
The solution, if pain persists - I can't say enough for the trigger point therapy, if it's persistent. But on a practical level. have your cab at your ears, so you hear well enough that you can play with a lighter touch all around - as well as not being as loud to the rest of the band, contributing to the whole band playing too loudly and killing both ears and hands. AND, take breaks to loosen your hands up every chance you get. It takes time to build stamina - any session, rehearsal, recording, personal practice - better to give the hands a rest every hour at least or so. I've worked with some cats who are so oblivious that they work in those sorts of things (not talking live shows) for 3-4 hours non-stop. That's just abusive.
If any of us want long careers or recreation playing, for the rest of our lives, it's critical to learn to relax and pace oneself - making that conscious effort, both physically and gear-technically to let the instruments and amps do the work and not to abuse our bodies in the process. (That means writing too much on the computer, too - LOL :-D - GUILTY GUILTY GUILTY )
I don't think I'm gripping the pick too tightly, but I hold it like this. Maybe it's the cause. But I don't know how to hold it any other way 
I just wan't to play with pick, that's all.
