For each k ∈ N, we describe a mapping $f_{k}:\mathbb{C} \longrightarrow E_{k}$ into a suitable non-complete complex locally convex space $E_{k}$ such that $f_{k}$ is $k$ times continuously complex differentiable (i.e., a $C^{k}_{\mathbb{C}}$-map) but not $C^{k+1}_{\mathbb{C}}$ and hence not complex analytic. We also describe a complex analytic map from $\ell^{1}$ to a suitable complete complex locally convex space $E$ which is unbounded on each non-empty open subset of $\ell^{1}$. Finally, we present a smooth map $\mathbb{R} \longrightarrow E$ into a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point.