 Solution

In general, no direct list fold with result type [a] can implement dropWhile p. Choose total values $u,v$ for which $p(u)=\mathrm{True}$ and $p(v)=\mathrm{False}$. The results on [] and [u] are both []. A hypothetical implementation foldr h z would therefore receive identical folded-tail values when extending either input by the same head $v$. Yet the required outputs are [v] and [v,u], respectively. This contradicts the fold factorisation criterion: the combining function cannot distinguish information already discarded by its tail summary.

The claim concerns a direct same-result-type list fold. A list fold returning extra information, followed by a projection, is possible: a pair can retain both the original tail and the dropped list suffix. Also, special predicates can have direct list folds: a constantly false predicate gives the identity; a constantly true predicate can use foldr (\_ r -> r) [], which correctly diverges when it must traverse an infinite spine. Those special cases do not supply an implementation for an arbitrary predicate.