How can I map a function to a list and stop when a condition is fulfilled and tell me if it stopped or reached the end?

I'd use span for this. It's like takeWhile but it gives you a pair with the remainder of the list as well as the matching part, like this:

> span (<3) [1,2,3,2,1]
([1,2],[3,2,1])

Then you can check if the remainder is empty:

haltmap :: (a -> Bool) -> (b -> a) -> [b] -> Either [a] [a]
haltmap p f xs = (if null rest then Right else Left) ys
  where
    (ys, rest) = span p (map f xs)