I carry a very basic first aid kit with me when I play (I have an IFAK on my kit... might as well use it!), saline for washing eyes if necessary, 2 israeli bandages, some sticky plasters, a tourniquet (you never know...), a small roll of gauze, and some zinc oxide tape. Covers all the injuries I'm competent to deal with myself in the short term ranging from a scuffed knee to an arterial bleed.
Most of the likely first aid problems at an airsoft game just don't have a 'first aid' solution that you can easily carry around, heart attacks, sprains, broken bones, eye trauma... doesn't matter what kit you have to hand with a lot of those, it's going to be bad!