FYI, in English (at the very least, in many regions of the world) "I'm fine" as used here is short for "I don't need it." Probably the raw Japanese was something like "ii desu / iinda".
She was just taking out her lip balm to casually show him that she had lip balm, not actually offer it to him. His overactive imagination caused him to jump to conclusions, and reject her imagined offer. She picked up on what he was thinking, and just teased him about it in good humor. (Because of course, a girl wouldn't offer to lend her lip balm to a guy, normally, unless she's clueless or she's deliberately being intimate, right? Indirect kiss awareness and all that...)