If pref(L) is regular, does that imply L is regular?
What you say doesn't make sense. How can you operate with a NFA for $L$ if you cant be sure that $L$ is regular?

Try to consider this language
$$L=\{ uu \mid u \in \Sigma^*\}$$
This language is not regular, it is not even context-free, which can be shown with the pumping lemma. I think from here you can continue by yourself. What is $\text{pref}(L)$?