That said, even as an IC, at higher levels the scope of impact you're expected to deliver at some point exceeds what one person working alone can have, so there's still lots of people leadership skills needed to advance: working cross-functionally and setting direction, communication, and ultimately influencing others to jump onboard and help you out.
(Note that this is different from people management -- e.g. formal coaching, performance reviews, dealing with HR/org/budget issues, etc.)