Try spray-on deodorant, that may help.
Don't cut it out. If you are planning on breastfeeding, it can damage your supply on that side. When your milk comes in it may become engorged, but not stimulating it and using a cold pack will cause it to go down.
It happened to me it should get better in a few days
Breast tissue does extend to the armpit.
If deodorant is painful either don't use it or you just dust a little baking soda in your armpit