I encountered this person (guy) who mischievously and annoyingly refers to another person (girl) as arrogant when that person is not. I suspect that the guy is trying to tease the girl but in doing so, he made himself appear as childish and I am trying to find a word to describe this kind of character.
Wind-up merchant — Wiktionary
noun (England, colloquial) One who enjoys winding others up in the sense of making fun of them or playing practical jokes
British colloquialism, but I would describe them as a wind up merchant (note it's wind as in wound, not wind as in north wind!)
Impish (comparative more impish, superlative most impish) Wiktionary
(Adjective) impish (comparative more impish, superlative most impish); mischievous; of or befitting an imp. Synonyms (naughtily or annoyingly playful): implike, mischievous, pixilated, prankish, puckish
Puckish is an interesting one. I didn't know that one before.
Perverse (comparative perverser, superlative perversest) Wiktionary
(Adjective) Turned aside; hence, specifically, turned away from the (morally) right; willfully erring; wicked; perverted.
Willfully perverse might work better.
Teaser — Wiktionary
noun One who engages in teasing; a tease.
This might have other unfortunate meanings if you did not use it carefully.