In a way that shows playful romantic interest or teasing behavior.
From 'flirting' (the present participle of 'flirt') plus the adverbial suffix '-ly', which comes from Old English 'lice' meaning 'in the manner of.'
The '-ly' suffix is one of English's most productive tools for making adverbs—nearly any adjective can become an adverb this way, which is why English speakers can instantly understand 'flirtingly' even if they've never heard it before.
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