1) Do you have an accessible attic where you can get up here and look at the underside of the roof sheathing aboveand aroundthe dormer ? If a main roof or flashing leak running down and exiting at the window, you should see water running down the underside of the sheathing, and staining. If none there, then that leaves you with the dormer roof and flashing, the window itself, or siding above the window, which are more likely unless you have a flat-topped dormer.
2) Since you say it occurred in snowy conditions, that should rule out a leak at the window itself unless it has a large projecting top brow ledge that can hold snow, and also siding issues unless you are missing entire pieces. Ditto if happens during snow melt but when there is no active precipitation falling. If you have a peaked or domed dormer roof, pretty much leaves dormer roof surface or the flashing around the dormer - and since coming in at top of window, sounds like roof.
3) Also, except in VERY cold weather, you would not get enough frost buildup in an attic, in general, to cause enough of a drip when melting to make it down the wall to the next floor ceiling. Also, this frost melting would occur during a thaw after HARD freeze spell (typically below about 10 degrees F), not during rain and warm snow conditions also. Pure condensation, without frost buildup to allow it to build up thick, rarely makes it past the point of molding insulation and forming mildew on attic wood, and VERY rarely gets to the point of flowing anywhere more than a few little drips trickling down the underside of the roof sheathing. Can certainly cause attic problems, but I have never heard of it (except from condensation on inside of window glass) causing this sort of significant wetting, and you say drips from top of window so certainly not that.
4) I am assuming you have a peaked roof - if flat-topped, then I would bet it is leaking on a flat part and running down the sheathing or roof underlayment to a point above the eaves, and the roofing is terminated wrong there so it is coming down through the roofing above the eaves into the wall.
5) Have you checked gutters over the dormer, if you have any, that they are not blocked up and getting backup into the roofing edge and down into the wall. Generally a dormer would not have gutters, but if you have a flat-topped dormer with gutters, then that interface would be highly suspect - more so than the possibility of a roof leak further up the roof migrating down and just happening to come out at the window. Worst possible case would probably be something like this, a flat or inverted V roof with a flat-topped dormer - like the 31st photo down as I count it, with this caption - This mansard roof on Central Street has a convex profile.-
6) A sure way to tell - I tend to be a "dig in there and find the darn source of the problem" kind of guy - since the drywall is damaged already, carefully with a utility knife remove damaged drywall right above the window frame, opening it up till you can see where the water is coming down - inside the wall (and on which face - inside drywall or inside the siding), or coming in from outside at top of window frame. That would be pretty definitive, and if coming from above the window you could open up a strip of drywall up the inside of the wall (assuming you do not have fancy interior walll treatment that cannot be matched) and "chase" the leak to the ceiling to see where it is coming down the wall. If you have fiberglass or blown-in insulation, you do not even have to open much of a hole - reach up in a small hole and pullthe insulationn abovge the window out, then use a strong flashlight and mirror to look up into the wall. Could be a leak right above the window, or if you have vapor barrier above your ceiling, could be a leak further upslope on the roof or in the flashing dripping down onto the vapor barrier, then flowing to the low spot and pooling till it overflows - which might just be above the window.
7) Please let us know what you find the problem to be - so we can improve our advice for other questioners down the road. Answering questions like this tends to be a bit of a ouija board operation - we try to avoid getting into a case of the blind leading the blind, but without a hands-on look-see or photos, can be tough to give accurate advice.