  In the present article, we extend previous results of the author and we show
that when $K$ is any quadratic imaginary field of class number one, Fermat's
equation $a^p+b^p+c^p=0$ does not have integral coprime solutions $a,b,c \in K
\setminus \{ 0 \}$ such that $2 \mid abc$ and $p \geq 19$ is prime. The results
are conjectural upon the veracity of a natural generalisation of Serre's
modularity conjecture.
