Go to in-game inbox, tick one or several boxes then click delete selected messages button either the top or bottom one and it will not delete the messages, like Sunny said, top button just refreshes the interface and bottom is non-functional.
Interesting. I still have 2 invisible wishes hidden in the WP, perhaps it's one of those, I can't check it to be sure because I don't want to waste a WP just to see it. I assume @Aia del Mana have already checked?