As per the question, I might have spelt cleanly wrong...
'Cleanly' meaning (of a person) 'habitually clean and careful to avoid dirt' is archaic, and should be used with care, e.g. in poetry or period dialogue.
Habitually clean and careful to avoid dirt.
‘some plain but cleanly country maid’
An adjective cleanly (pronounced /ˈklɛnli/, unlike the adverb /ˈkliːnli/, but spelled the same way as the adverb) certainly exists. This is documented by various dictionaries, including the free online Merriam-Webster Dictionary, American Heritage Dictionary, Oxford "Living" Dictionaries, and Collins Dictionary. (I found all these entries through the OneLook Dictionary Search, a useful tool for researching words that you are not sure about.) The word "habitually" seems to be a common element of the definitions given by these sources.
Although it "is a word" in the sense of being in the dictionary, to my ears, cleanly as an adjective sounds fairly odd. I'm not sure I would go so far as to call it "archaic", as the Oxford Living Dictionaries entry quoted by Michael Harvey's answer says, but I agree with that answer's recommendation to use the word with care. I can't think of a situation where it would come naturally to me.
The adjective cleanly is less frequent today than it was a century ago
The adjective cleanly seems to be used less often nowadays than it was in the past. I looked on the Google Ngram Viewer for the top matches to "a cleanly *" (which wouldn't necessarily all have adjective cleanly, but I thought that many would) and basically all of them went down a lot in frequency over the course of the 20th century:
In a comment, Nathan M. mentioned the OED entry for this word, but I don't think that's very useful for getting a picture of how or whether this word is used today because the entry was first published 1889 and has not yet been fully updated. The most recent OED citation for sense 3a, "Of persons (or beasts): Addicted to cleanness, habitually clean; careful to avoid filth," is from 1885.
Recent examples of the adjective cleanly
Cleanly as an adjective is not entirely absent from more recent texts. I found an example of it being used in an article from 1993:
One might recall in this connection Leonardo's somewhat different paragone, that of the grimy toil of the sculptor with the cleanly activity of the painter.
(p. 585, "Concerning the 'Mechanical' Parts of Painting and the Artistic Culture of Seventeenth-Century France", by Donald Posner, in The Art Bulletin December 1993 Volume LXXV Number 4)
More examples are mentioned in the quotes in the following section.
Discussion of the adjective cleanly in usage guides etc.
I found the adjective cleanly mentioned in Robert Allen's Pocket Fowler's Modern English Usage, 2nd edition (2008) with no special warning about its use, although Allen does imply that the reader may not be particularly used to hearing it:
the adjective cleanly (pronounced klen-li) [...] means 'habitually clean': persons of refined and cleanly habits and decent language—Spartacus International, 2003. This meaning will probably be more familiar in its derivative noun form cleanliness, which is proverbially next to godliness.
Bryan Garner's Modern English Usage (2016) says
sometimes cleanly functions as an adjective—and is pronounced /klen-li/—in a sense corresponding to the noun cleanliness. It means either (1) "(of a person) habitually clean"; or (2) "(of a place) habitually kept clean." In sense 2, a simple clean is surely preferable. In the first and second examples that follow, sense 1 applies. In the third, sense 2 applies:
"'Owing to the leaning and handling of dirty persons, tobacco-spitting, the deposit of broken fruit and waste of all sorts of eatables, and other filthy practices voluntary or otherwise, the summer houses, seats, balustrades, balconies of the bridges are frequently forbidding to cleanly persons, who are thus deprived of what they deem their rights upon the Park.'" Elizabeth Barlow, "Rebuilding the Olmstead," N.Y. Times, 9 May 1981, § 1, at 23 (quoting Frederick Law Olmstead, one of Central Park's designers and its original administrator, from a writing dated 1860).
"'The people who park here are very cleanly people. They don't leave any cans or bottles,' he said." Estella Villanueva, "Fair Neighborhood Fares Well," Des Moines Register, 28 Aug. 1996, ATE §, at 1 (quoting Bob Wilcox).
"'Our whole approach to quality assurance is not cracking the whip but to point out why things like dusting the pictures, a cleanly [read clean] room, are important," [Ray] Sawyer said." Timothy N. Troy, "Budget Host Cultivates Quality," Hotel and Hotel Mgmt., 15 Aug. 1994, at 3.
Note that the first example is antique and that the second and third examples occur in reported speech. Today cleanly is more common in speech than in writing.
Yes, you can use cleanly. You can use it as either an adverb or an adjective. Merriam-Webster provides one adverbial definition and two adjectival definitions for cleanly.
- Examples from Dictionary.com:
adjective, clean·li·er, clean·li·est.
personally neat; careful to keep or make clean: The cat is by nature a cleanly animal.
habitually kept clean.
Obsolete. cleansing; making clean.
- in a clean manner.
Examples from Cambridge Dictionary:
cleanly adverb (honestly)
fairly and honestly: The election campaign was not conducted very cleanly.
cleanly adverb (not roughly)
with smooth straight edges: The plate broke cleanly in half.
cleanly adverb (equally)
equally: Opinions were split cleanly between men and women.
Examples from Merriam-Webster of cleanly in a Sentence:
A sharp knife will cut through the skin of a tomato cleanly.
This fuel burns more cleanly than other fuels.
Recent Examples of cleanly from the Web
Gaining control of her hands and fingers eventually allowed her to twist one piece into the correct position and snap it cleanly into place. - Aaron Gilbreath, Longreads, "The Inward Empire," 27 June 2018
Turn carefully using a metal spatula to cleanly lift off grate and grill until lightly charred on second side and warmed through, about 2 minutes more. - Chris Morocco, Bon Appetit, "Easiest-Ever Grilled Veggie Burgers," 25 June 2018
Astros left fielder Marwin Gonzalez fielded the ball cleanly and made a terrific one-hop throw to catcher Brian McCann. - Blair kerkhoff, Kansascity, "Ump John Tumpane ejects Mike Moustakas ... again," 17 June 2018