My guitar teacher is showing me some ballad chord progressions (while many are widely used) and playing notes within the chords instead of just strumming. Is there some unique way of composing said genre?

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Chord progressions aren't anywhere near as important to ballad writing, as is the tone, instrumentation, tempo, rhythm etc.

"People Ain't No Good" by Nick Cave - uses the most generic progression imaginable (C - G - F) that has been used in countless rock songs. Yet it still sounds like a ballad, so obviously the chord progression isn't that important.
So he's playing chords and arpeggios? And you're.... yah, I'm lost.

I don't think I understand your question at all.

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This thread might be relevant to the topic being discussed. They're jazz cats (i.e. really dry) but they know their ****.

TLDR: There's no "technique" that automatically renders a ballad. It's more something to do with the feelings behind it, you know. Tender, lovey-dovey stuff. All the other superfluous artifacts that often come along with ballads (Medium-slow tempo, lyrical melodies, relaxed harmonic rhythm) do not define a ballad, but reflect the form/inspiration.
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