**Abstract:** We use Oda's definition of the Schwarzian derivative for locally univalent holomorphic maps F in several complex variables to define a Schwarzian derivative operator phi F. We use the Bergman metric to define a norm 11,vertical bar vertical bar phi F vertical bar vertical bar for this operator, which in the ball is invariant under composition with automorphisms. We study the linearly invariant family F-alpha = {F : B-n -> C-n vertical bar F(0) = 0, DF(0) = Id, vertical bar vertical bar phi F vertical bar vertical bar <= alpha}, estimating its order and norm order.