Suppose you have a differentiable map $\Phi : E \rightarrow F$ where $E$ and $F$ are Banach spaces, and a curve $t \mapsto u(t)$ of elements of $E$, with $u(0)=0$ and $\Phi(u(t))$ constant, such that $\frac{u(t)}{t}$ converges weakly-* in $E$ to a certain $v$ (or possibly even a weaker assumption : there is a certain sequence of times $t_n$ such that $\frac{u(t_n)}{t_n}$ converges weakly-* to $v$ in $E$).
Does this imply that $v \in \ker D\Phi(0)$ ?