It depends on what you mean by "has an exact form". If you mean "has a single, invariant form", the answer would most likely be "no". According to several definitions of "stem", Latin words (including nouns) may have more than one stem (or alternatively, you could say that the stem of a word may have more than one form: it's a bit difficult to distinguish these two possibilities). The use of different stems for different forms of the word is called "heteroclisy".
Different definitions of "stem"
The concept of a "stem" is not completely clear/unambiguous. I'm aware of at least three distinct definitions:
Etymological/historical linguistics/comparative IE linguistics: The stem of līber in a historical sense is lībero-; the nominative singular form is historically derived from lībero-s via syncope and loss of s after r (as explained in the "Type A." section of Alex B.'s answer to Why do some 2nd decl. "-er" adjectives and nouns drop the "e" in the stem?).
This word happens to be built on a single stem/single form in a historical sense, but that is not true for all words. IE "heteroclite" nouns existed; an example of a descendant in Latin is iecur "liver". This type of heteroclite noun is mentioned in the answers to Does any Latin noun originally end in -r?
Synchronic analysis of Latin morphophonology according to certain modern linguistic theories: The stem of līber is a bit difficult to analyze; one option would be to treat this as a synchronically heteroclite adjective in Classical Latin, with the masculine nominative being built on a consonant-final stem līber- and the other masculine forms being built on a vowel-final stem lībero-. This seems to be the analysis that would be adopted by Cser 2016, who refers to the masculine nominative singular piger as being built on a stem pigr-, with no stem-final vowel (p. 129), despite the use of [ō] in the ablative singular form pigrō.
Draconis's answer indicates that "stem" has been used in some sources to refer to a different concept that is similar to what the other two approaches would call the "root" or "base" of a noun. I'm not sure which specific sources use that definition. It might have lasted longer in pedagogical contexts than in linguistic texts.
Heteroclisy in more detail
One clear example of a heteroclite Latin noun is vās: the ablative singular vāse is built on the stem vās-, while the genitive plural vāsōrum is built on the stem vāso-. (Historically, the explanation in this case is that the different stem is derived from a "collateral" form of the noun; Lewis and Short says that in anteclassical Latin, vāsum existed as a separate nominative singular form. But in Classical Latin, the stems vās- and vāsō- came to be used in complementary contexts, which allows vāse and vāsōrum to be analyzed as forms of a single noun. I've generally seen this noun described as having the stem vās- in the singular and vāso- in the plural, although the nominative/accusative plural vāsa is formally ambiguous: it would look the same regardless of whether it represented vās- + -a or vāso- + -a).
A number of other Latin nouns may or may not be analyzed as "heteroclite" depending on how you think of the idea of a "stem". This issue came up in a discussion I had with Joonas Ilmavirta beneath the answer that he posted to the question What consonants can a noun stem end in? Many third-declension nouns show an i in the genitive plural (before the ending -um) but not in most of the other forms (e.g. vermis has the genitive plural vermium, which seems to be built on the stem vermi-, but the accusative vermem and the ablative verme, which seem to be built on the stem verm-).
The stems of nominative singular forms
The situation gets even more complicated if you try to include nominative singular forms in your analysis; for this reason, I've seen a number of pedagogical texts that don't attempt to relate the nominative singular form to a stem, but that just present the nominative singular as a form that must be memorized as a whole. Nonetheless, from an etymological and possibly even from a synchronic perspective, almost all Latin nominative singular forms can in fact be analyzed as being composed of a stem and a suffix.
According to the theories that use definitions 1 or 2 of the term "stem", Latin has a nominative singular suffix with a fairly small number of variant forms or allomorphs: mainly -s, -∅, and -m. (Cser 2016 lists one additional allomorph, -ēs, to account for certain third-declension nouns (p. 127)). The form of this suffix is conditioned morphologically by the gender of the noun, and conditioned phonologically by the form of the end of the noun's stem (for example, nouns with a stem ending in a consistently take -∅).
Note that I did not include -r in my list of forms of the nominative singular suffix. As far as I know, Latin has no words ending in -er that would be analyzed as having a stem ending in -e. If you're familiar with that fact, you can determine that līber does not have the stem lībe.
For some words, the stem of nominative singular can be analyzed as being the same in an "underlying" way as the stem of the other forms, even though it appears different on the "surface". For example, the nominative singular form rēx be seen as being built synchronically on the same stem rēg- as the other forms of this noun. The explanation is that rēg- + -s becomes rēx according to known rules for combining Latin sounds: g turns into k before s.
I remember encountering theories that take a different approach to explaining variant phonetic forms like this (I forget where, and which language was being discussed): we could say instead that the stem of the noun rēx is "stored with" two allomorphs, rēg and rēk, and that there are rules that take these two allomorphs as an input and select the appropriate allomorph (rather than taking rēg as an input and turning its g into a k). This is a bit of a hairsplitting distinction in most contexts, however. Cser 2016 says that the "rule-based" framework adopted in that dissertation is simply for convenience.
The "stem" (def 1 or 2) is viewed as including the "theme vowel"
According to definitions 1 or 2, the "stem" of a noun includes the "theme vowel"/"thematic vowel". For example, servus is an "o-stem" noun, so the stem would end in o: servo-. (If I remember correctly, from an etymological perspective, I think only second-declension nouns have stems that end in a PIE "theme vowel"; the first-declension a-stem nouns and fourth-declension u-stem nouns come from PIE forms that ended in a consonant that was vocalized in the history of Latin). Allen and Greenough's grammar, which seems to use a very etymology-based definition of "stem", says that servus has the "base" serv- and the "stem" servo- (27a).
The form we get when we truncate the vowel (serv-) would be called the "root".
According to definition 2, Latin stems in this sense do show some allomorphy: for example, in the nominative singular form of servus we find [servu] (with short u) rather than [servo]. This is explainable in terms of diachronic sound changes, but the correct synchronic analysis isn't necessarily the same as the diachronic explanation. The stem vowel may (appear to) disappear entirely in some forms, such as the ablative/dative plural servīs. One possible synchronic analysis (given by Cser 2016) is that servīs is divisible into a vowel-final stem servo- and a vowel-initial suffix -īs, with a morphophonological process of vowel deletion resulting in the lack of [o] or [ō] in the surface form servīs. If I remember correctly, the historical development is a bit different: I think that the ī in forms like this comes from earlier diphthongs oi (or for a-stem nouns, ai), which were weakened outside of the first syllable to ei, which later monophthongized to [ī].
A similar analysis is often used in the linguistic analysis of Romance languages, the modern descendants of Latin. You can see an overview of this approach applied to Spanish in Bermudez-Otero 2012.
Cser, András. 2016. "Aspects of the phonology and morphology of Classical Latin".
Bermudez-Otero, Ricardo. 2012. "The Spanish lexicon stores stems with theme vowels, not roots with inflectional class features".