Let $\pi$ be a uniformizer of $F$, so 
$$
F = \pi^{\mathbf Z}(1+\mathfrak m_F) \cong \mathbf Z \times (1+\mathfrak m_F)
$$
as topological groups, where $\mathbf Z$ has the discrete topology. Thus your question is the same as asking if all homomorphisms $1+\mathfrak m_F \to \mathbf C^\times$ are continuous.

Now take $F = \mathbf Q_p$. Using the $p$-adic exponential and logarithm, $1+p\mathbf Z_p \cong p\mathbf Z_p$ as topological groups when $p\not=2$ and 
$$
1+2\mathbf Z_2 \cong \{\pm 1\} \times (1+4\mathbf Z_2) \cong \{\pm 1\} \times 4\mathbf Z_2.
$$
Since $p\mathbf Z_p \cong \mathbf Z_p$ and $4\mathbf Z_2 \cong \mathbf Z_2$ as topological groups, your question when $F = \mathbf Q_p$ for any prime $p$ is equivalent to asking if every group homomorphism 
$$
\mathbf Z_p \to \mathbf C^\times
$$ 
is continuous. Do you agree that there are a *lot* of discontinuous homomorphisms of that kind? 

Explicitly, because $\mathbf C^\times$ is a divisible group, Zorn’s lemma tells us each group homomorphism $\mathbf Z\to \mathbf C^\times$ extends (somehow) to a group homomorphism $\mathbf Z_p\to\mathbf C^\times$. So start with the mapping $\mathbf Z\to \mathbf C^\times$ where $n\mapsto 2^n$ and extend it to a homomorphism on $\mathbf Z_p$. That extension is not continuous since its image is unbounded and thus not compact.