It is a standard linear algebra result that a vector space $V$ over a field $\mathbb{K}$ is isomorphic to its dual $V^{\vee}$, but non canonically in general. If $V = \mathbb{K}$ then we have a canonical isomorphism:
$$ \require{AMScd} \begin{CD} \mathbb{K}^{\vee} @>>> \mathbb{K} \\ \\ \lambda @>>> \lambda(1) \end{CD} $$
My question is: can this argument be extended to the vector spaces of dimension one?
A vector space of dimension one is isomorphic to its base field, but not canonically. When trying to extend your argument to a vector space of dimension one, which vector $v\in V$ should take the place of the $1$ in $$\Bbb{K}^{\vee}\ \longrightarrow\ \Bbb{K}:\ \lambda\ \longmapsto\ \lambda(1)?$$ This comes down to choosing a basis for $V$, which is what makes the isomorphism non-canonical just as for higher-dimensional vector spaces.