If it is your left arm/hand, and you also notice it when playing guitar, you may be holding the upper part of your left arm too close to the body when playing.  It can cut off some of the circulation to the arm and hand when you do that.  I have found moving the arm out even a little bit helps. 

Sometimes the shirt or style/size of shirt you wear when playing also makes a difference.  The way the armpit of the shirt is cut.  Ever notice that some seriously scrawny guitar players sometimes are wearing oversize t-shirts and etc when playing?  Maybe sometimes it is fashion, but sometimes it is definitely to have a looser fit around the arm.

I run into that problem sometimes when playing too, StoneDog.  That's what works for me.  That and remembering to stretch and shake out the arm/hand once in a while between songs, but that is good practice to help keep tension from building up anyway.  Give it a try.  Hope it works for ya.

Fun is fun.  But all kidding aside, using alcohol to relieve pain or discomfort isn't really usually the best idea. It is numbness of it's own sort, and letting it mask something like numbness or pain rather than looking for a cause and dealing with it could result in permanent damage.  

Sorry to be the wet blanket on humor, but I felt it best if somebody mentioned that.