No & Yes.
No. Because unlike Props, there isn't a Pivot Point Selection Box.
A Character's Default Pivot Point is at a Character's  Bone Root, which is a virtual fixed Point at Position X, in Screenshot. It should be at a Characters Hips.
 
Yes. Forum Member Rampa produced a Tutorial for the Hip Bone as a Pivot Point. Send Rampa a Private Message.

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Can i change pivot position of character?
ex) center(default position) → left foot or right foot
This is necessary for character animation. you know..