Portable soldering sponge?
The most compact solution I've seen for this is sistema's 200ml rectangular KLIP IT containers. At 4.5"x3.5"x1.5" they push the boundary of "compact", but on the other hand they can take a decent-sized sponge. And the fact that they're water-sealed means that you can moisten the sponge where there's water and take it out onto a dry field without having to worry about the rest of your gear getting wet.