I've heard of languages that contrast [r] and [ɾ] but I am unable to find any language that contrast a normal trill and a long trill. I searched far and near but to no avail. So is there any language that contrast [r] and [r:]?
(Due to finding a source that I think provides an even better example than my previous answer, I've updated this post).
In “Initial and Medial Geminate Trills in Arop-Lokep,” by Raymond, Mary, and Steve Parker (Journal of the International Phonetic Association, vol. 35, no. 1, 2005, pp. 99–111. JSTOR, www.jstor.org/stable/44526399. Accessed 17 July 2021.), the authors describe a contrast in the Austronesian language Arop-Lokep between a singleton trill /r/ [r] and geminate trill /rr/ [rr] in word-initial and intervocalic position.
They say that historically, word-initial geminates in Arop-Lokep come from reduction of an intervening vowel in forms with word-initial reduplication (page 101).
Word-initial trills always have more than one contact (page 103), with a mean of 2.95 contacts for word-intial singleton #rV and 5.00 contacts for word-initial geminate #rrV (page 105, table 3).
Intervocalically, both [rr] and [r] contrast with [ɾ], as [ɾ] occurs as an allophone of intervocalic /d/ in unstressed syllables (page 99). Intervocalic [rr] may optionally degeminate to [r] in many words (page 102).
Original answer below:
It's very common for a non-geminate rhotic /r/ to have a possible realization as a tap or flap [ɾ]. However, in many cases a trill realization is not absolutely excluded.
A contrast between short and long trills could occur in languages where singleton /r/ is described as having a potential trill allophone, and where singleton /r/ contrasts with geminate /rr/. The first candidate I thought of is Italian; I thought it was also possible this might be the case for certain Slavic languages such as Russian, but after looking into it more, they seem less likely to be good examples.
Many varieties of Italian have a relatively firm length contrast for most consonants, including /r/ vs. /rr/. Although length contrast is much more marginal in Slavic languages, many of them have contrastive consonant length to some degree. Based on my research so far, it appears that a length contrast between /r/ and /rr/ is even more likely to be marginal than a length contrast for other consonants, so a Slavic language is probably not a good answer to your question (but I'll leave the mention of Russian in this answer since I originally included it).
"Rhotic Variation in Tuscan Italian", by C. Celata, A. Vietti, and L. Spreafico says that Ladefoged and Maddieson describe contemporary standard Italian as having one or two apical contacts for intervocalic singleton /r/ compared to "no less than five and up to seven" contacts for intervocalic geminate /rr/. Celata, Vietti, and Spreafico say that a pronunciation of singleton /r/ with a single contact is categorized by Ladefoged and Maddieson as a trill rather than as a tap or flap when the consonant is produced with the articulators in the same position used when vibration occurs (Romance Phonetics and Phonology, page 92).
The realization of rhotics in different varieties of Italian appears to show some variety so unfortunately these facts may be hard to corroborate.
Some Slavic languages (but gemination might be of dubious status)
Russian's two rhotic consonants (palatalized and non-palatalized) are both generally described as trills, although non-trill allophones are possible. Geminate [rr] is "quite unlikely" ("A Reference Grammar of Russian", by Alan Timberlake, page 68) and never occurs word-internally in native vocabulary, but I believe /rr/ can occur in the pronunciation of words of foreign origin, although there is likely very little if any effective contrast with singleton /r/. I am not sure what specific examples there would be of /rr/ (spelling is no clear guide, since there are words spelled with "рр" and pronounced with single /r/) but according to Olga Dmitrieva,
In present-day Russian any consonant can be geminated (and pronounced as phonetically long) with an exception of the palatal glide [j]
("Geminate Typology and the Perception of Consonant Duration", 2012, page 60).
Dmitrieva notes that the status of Russian geminates is disputed and they can be subject to processes of degemination (apparently liquids are especially prone to degemination in Russian).
I believe a similar situation occurs in a number of other Slavic languages.
According to Wikipedia, Slovak does contrast /r/ and /rː/, but the Journal of IPA on Slovak says that the geminated /r/ (i.e. /rː/) is considered an allophone of /r/ because there are no minimal pairs of /r/ and /rː/. They're not in complementary distribution: the geminated /r/ ‘can only occur in the syllable nucleus’, the short /r/ ‘in the nucleus and elsewhere’. It further says that the /r/ is mostly realised as a tap [ɾ].
Also, as @Jafe pointed out, Finnish contrasts [r] and [rː]. Another minimal pair would be:
- käry - [kæry] (‘smell of burning’)
- kärry - [kærːy] (‘cart’)