  In this paper, authors show the ill-posedness of 3D incompressible
Navier-Stokes equations in the critical Triebel-Lizorkin spaces $
\dot{F}^{-1,q}_{\infty} (\mathbb{R}^3) $ for any $ q>2 $ in the sense that
arbitrarily small initial data of $ \dot{F}^{-1,q}_{\infty}(\mathbb{R}^3) $ can
lead the corresponding solution to become arbitrarily large after an
arbitrarily short time. In view of the well-posedness of 3D-incompressible
Navier-Stokes equations in $ BMO^{-1} $ (i.e. the Triebel-Lizorkin space $
\dot{F}^{-1,2}_{\infty}(\mathbb{R}^3) $) by Koch and Tataru, our work completes
a dichotomy of well-posedness and ill-posedness in the Triebel-Lizorkin space
framework depending on $ q=2 $ or $ q>2 $.
