Yes, it is possible to drop your weapon, though as noted in the question, there is *not* a dedicated button for it. After searching for a while, I found [this thread](https://www.gamefaqs.com/boards/124913-killing-floor-2/74632843) that explains how to do it:

>You have to be holding the weapon, hold triangle and hit R1

Thus, to drop a weapon, equip it, hold triangle, and press R1.

[![image gun menu][1]][1]

It's worth noting there are some weapons that you *cannot* drop, such as the 9mm and your knife. These do not take up any weight, so it will not affect what you can carry, though.


  [1]: https://i.sstatic.net/MCuXw.jpg