Is implementing this words function possible without a postprocessing step after folding?

If I understand correctly, your requirements include

(1) words "a b c" == words " a b c" == ["a", "b", "c"]
(2) words "xa b c" == ["xa", "b", "c"] /= ["x", "a", "b", "c"] == words "x a b c"

This implies that we can not have

words = foldr step base

for any step and base.

Indeed, if we had that, then

words "xa b c"
= def words and foldr
step 'x' (words "a b c")
= (1)
step 'x' (words " a b c")
= def words and foldr
words "x a b c"

and this contradicts (2).

You definitely need some post-processing after the foldr.

@chi has a wonderful argument that you cannot implement words using "a" fold, but you did say using folds.

words = filterNull . words1
    filterNull = foldr (\xs -> if null xs then id else (xs:)) []
    words1 = foldr (\c -> if c == ' ' then ([]:) else consHead c) []
    consHead c []       = [[c]]
    consHead c (xs:xss) = (c:xs):xss

Both the outermost and innermost function are folds. ;-)

Yes. Eventhough it's a little tricky you may still do this job properly by using a single foldr and nothing else if you dwell into CPS (Continuation Passing Style). I had shown a special kind of chunksOf function previously.

In this kinds of folds our accumulator, hence the result of the fold is a function and we have to apply it to an identity kind of input so that we have the final result. So this may count as a final processing stage or not since we are using a single fold here and the type of it includes the function. Open to debate :)

ws :: String -> [String]
ws str = foldr go sf str $ ""
         sf :: String -> [String]
         sf s = if s == " " then [""] else [s]
         go :: Char -> (String -> [String]) -> (String -> [String])
         go c f = \pc -> let (s:ss) = f [c]
                         in case pc of
                            ""        -> dropWhile (== "") (s:ss)
                            otherwise -> case (pc == " ", s == "") of
                                         (True, False)  -> "":s:ss
                                         (True, True)   -> s:ss
                                         otherwise      -> (pc++s):ss

λ> ws "   a  b    c   "

sf : The initial function value to start with.

go : The iterator function

We are actually not fully utilizing the power of the CPS here since we have both the previous character pc and the currect character c at hand in every turn. It was very useful in the chunksOf function mentioned above while chunking a [Int] into [[Int]] every time an ascending sequence of elements were broken.