  We prove the ill-posedness for the 3D incompressible inhomogeneous
Navier-stokes equations in critical Besov space. In particular, a norm
inflation happens in finite time with the initial data satisfying
$$\|a_0\|_{\dot{B}_{p,1}^\frac{3}{p}}+\|u_0\|_{\dot{B}_{6,1}^{-\frac{1}{2}}}\le
\delta,\ p>6$$ or
$$\|a_0\|_{\dot{B}_{6,1}^\frac{1}{2}}+\|u_0\|_{\dot{B}_{p,1}^{\frac{3}{p}-1}}\le
\delta,\ p>6.$$ To obtain the norm inflation, we construct a special class of
initial data and introduce a modified pressure. Comparing with the classical
Navier-Stokes equations in $L^\infty$ framework, we can obtain the
ill-posedness for the inhomogeneous case in near $L^6$ framework.
