Is this "ice damming"?
Even if you are absolutely sure there is no water leaking into the attic you still have a problem. I am guessing that the water is getting pushed up and under the drip edge. It is migrating under the soffit trim and dripping out where it meets the siding. Potentially the water may get behind the siding if it builds up far enough. There are a variety of methods for dealing with the ice. Try to remove as much of the snow as you safely can. Melt the ice with an ice melt product that doesn't contain rock salt. If the roof is higher than you are comfortable working on you can buy ice melt in a block form that can be thrown onto the roof from the ground. You can also throw pantyhose filled with ice melt onto the roof. All the ice doesn't need to be removed you just need several channels that run from the eave toward the ridge to allow the water to drain from under the ice.