(edit: changed title as per Cev’s recommendation; there ought to be a standard term for these)
Some languages have contrastive phonemes that appear in only one root: for example, Norwegian /ʉi̯/ only appears in the word /hʉi̯/ hui and the derived verb huie; Arabic /lˤ/ only appears in /alˤlˤah/ ‘Allah’; Spanish /ui̯/ only appears in muy; and in standard American English, unconditioned /eə/ only occurs in the interjection ‘yeah’. (Interjections, of course, often have expanded phoneme inventories relative to the rest of the language—see English oink and boink, which have /oi/ before a non-coronal cluster—but these other two examples are of ordinary roots.)
According to Wikipedia, Dahalo has five such phonemes: /ⁿd̠ʷ/, /ᶮdʒ/ in /kípuᶮdʒu/ ‘place where maize is seasoned’, /ᵑɡʷ/ in /háᵑɡʷaraᵑɡʷára/ ‘centipede’, /ɬʷ/, in /ɬʷaʜ-/ ‘to pinch’, and /j/, in /jáːjo/ ‘mother’.
According to Blust’s Austronesian Languages, unconditioned nasal vowels only appear in one root in Bintulu and Miri: Bintulu has the negation marker [ʔã] ã and Miri has the minimal pair haaw ‘rafter’ : hããw ‘2sg’, but no other unconditioned nasal vowels are known in either language.
Mako /ə/ only appears in the past-tense suffix -tə, but it’s fully contrastive: every vowel can appear word-finally after /t/.
Qiang /ɦ/ only occurs in the interjection /ɦa/ and a directional prefix. (There are many other hapax phonoumena in the Qiangic languages; I won’t list them all.)
Hoyahoya /s/ only occurs in /sa/ sha ‘bone’. (/s/ is written sh because the other fricatives, /ʁ/ and /h/, are written gh and h.)
The Amuzgo syllabic velarized prenasalized bilabial trill only occurs in the word [ʃa˥m̩ˠʙˠ˥] ša1ṃb1 ‘antlion’.
Mianchi /ɬ/ occurs only in /ɬə̀/ ‘moon’ ( < PTB *s-la), and only for some speakers; others have /l/.
In some dialects of Mandarin, the final -iai only occurs in 崖 yái ‘precipice’.
Sometimes these phonemes are supported by loanwords.
Japhug /y/ occurs in Chinese loanwords, but only in one native root, /qaɟy/ ‘fish’.
Yadu /æ/ occurs in Chinese loanwords and one native root, /tsæm/ ‘girl’.
In Longxi, /h/ occurs in two loanwords, an onomatopoeic form, and /hàN háN/ ‘corridor’; and /v/ occurs only in words derived from Proto-Qiang *u ‘you’: /vù/ ‘you’ and its compounds, /vú lià/ ‘we (incl.)’ and its dual, and /vé ì/ ‘yourself’ and its dual and plural; however, /v/ may also be analyzed as an allophone of /u/ in word-initial position (pure [u] never appears word-initially), making these forms /ù/, /ú lià/, and /ué ì/, with a u-deletion rule for the reflexive.
Sierra Miwok /š/ occurs only in English loanwords and the exclamation /ʔiš·o·/ ‘Scat!’.
Sometimes they occur in several roots. Chechen /r̥/ only occurs in /vuor̥/ ‘seven’ and /bar̥/ ‘eight’. The closely related language Bats has /vorɬ/ and /barɬ/.
Sometimes they are supported by the morphology. In Finnish, the diphthong ey merged into öy except in the verb leyhyä, but it can still occur in derived words, when e- is adjacent to -U in a front-harmonic context.