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.

Since $\mathbf C^\times$ is a divisible group, Zorn’s lemma tells us for an abelian group $G$ and subgroup $H$ that each group homomorphism $H\to \mathbf C^\times$ extends (somehow) to a group homomorphism $G\to\mathbf C^\times$. Use $G = 1+\mathfrak m_F$ and $H = \langle 1+\pi\rangle$, which is isomorphic to $\mathbf Z$. Start with the homomorphism $H\to \mathbf C^\times$ where $(1+\pi)^k \to 2^k$ and extend to a homomorphism with domain $1+\mathfrak m_F$ by Zorn’s lemma. This homomorphism is not continuous since $1+\mathfrak m_F$ is compact but the homomorphism on this is discontinuous because its image in $\mathbf C^\times$ is unbounded and thus not compact.