Ok, I finally had a pet that could take on one of the 'boss' creatures. It says I won, but if I try to close out of it I get the warning: "You will leave the island if you give up. Do you really want to give up?" If I won, shouldn't I just be able to exit the battle and keep exploring?

Never mind, I think it was just a minor glitch. I closed the tab and opened an Explore in another tab, and it said I got the items. Very odd, though.