Take for example the equation
$\dot x = \lambda x$
$\dot y = y^2$
For $\lambda < 0$. The origin is an unstable equilibrium point, however, its stable manifold is the whole lower semiplane (including the $y=0$ axis).
If there is one eigenvalue of modulus bigger than $0$ for the derivative in the equilibrium point, the result is true. See for example here.