Readers might have noticed that I am working on the availability of Buddhist texts after the disappearance of Buddhist communities in South India. Did the vanished Buddhist communities leave beyond libraries of Buddhist texts? —I have no evidence of that. Did Jainas collect Buddhist texts also in South India?

The latter possibility seems to me more likely. This led me to some investigation on Jaina libraries in India in general.
An interesting article is The Jain Knowledge Warehouses: Traditional Libraries in India, by John E. Cort (published on the Journal of the American Oriental Society, Vol. 115, No. 1 (Jan. – Mar., 1995), pp. 77-87, available on Jstor). Cort deals primarily with the Jaina library in Pāṭaṇ (Gujarat) and has collected a lot of historical information, especially on its last centuries. However, I have collected and re-arranged here some general passages.
A first group of passages deals with the rationale of Jaina collections. Cort starts with the reason for collecting Jaina books:

Written copies of manuscripts have long played an important role in Jain intellectual, ritual, and community life. In the absence of any living enlightened teachers—according to Jain cosmological doctrines, enlightenment in this era became impossible shortly after the demise and liberation of Mahāvīra […]—the texts containing the teachings of Mahāvīra are essential for the guidance of the Jain community. (p. 77)

The next step is the reason for collecting also non-Jaina books:

Jains insist that a book, any book, should be treated with respect. Once a year, therefore, on the fifth day after the New Year, known as “Knowledge Fifth” (Jñān Pañcamī), Jains of to the libraries and bhaṇḍārs to worship both the knowledge contained in the manuscripts and the physical manuscripts themselves. Both modern printed books and older hand-written manuscripts are arranged in tiers on tables. Laity stand before the books with hands joined in a gesture of veneration, and sing vernacular hymns to Knowledge. Offerings of the sacred, charged sandalwood powder known as vāskep (as well as money) are made onto metal trays on the tables, and then, in an act sure to run shivers up the spine of any library archivist, the powder is sprinkled over the books and manuscripts themselves.
The very book and manuscripts as physical objects are to be treated with respect and veneration, and disrespect is considered as an aśātnā, or moral fault. (p. 87)

Beside that, Cort has some scattered passages on the history of Jaina libraries, from its beginnings:

Among the key events in the crystallisation of the split between the Śvetāmbara and Digambara sects were three Śvetāmbara councils held in Valabhi in Gujarat and Mathurā in north India in the fourth and fifth centuries to commit to writing standard editions of key Jain texts. According to a Śvetāmbara Jain tradition, the first libraries were built in the late eight century. (p. 78)

To its end:

The history of theses collections further reveals a dramatic change that has occurred in the last one hundred years, as Western notions of public libraries and research institutions have come to dominance in India (p. 86)

Cort also adds that manuscripts are now much less significant, given the wide availability of print.

Now, to the real possibility of Jaina libraries to survive across India:

Arranging for manuscripts to be copied for monks to use and establishing places for them to be kept were among the duties expected of laity as part of their support for and devotion to the monastic community. The three most important ‘fields of donation’ for medieval Śvetāmbara laity were images of the Jinas, temples containing such images, and Jain texts. Furthermore, the colophons on some manuscripts indicate that commissioning the copying of a manuscript generated merit that could be dedicated to a living or deceased ancestor. […] It is therefore not surprising that medieval Jain kinds and merchants were famous for the libraries that they established. (p. 78)

Thus, it seems that a living Śvetāmbara lay community would be a guarantee for the possibility of a library to be established and preserved. In case you are interested in the physical place, Cort has something for you:

The libraries themselves were kept either in small, dark, unventilated cellars, or in similar chambers above ground. (p. 79) […] Many Jain pilgrimage shrines still have secret cellars where, in times of political instability, images, ornaments, manuscripts, and other valuables could be stored for safe-keeping. (p. 80)

Up until the early decades of the twentieth century, the actual ownership of many of the manuscript collections was in the hands of specific mendicants who resided permanently in their monasteries. These mendicants, known as yatis, did not take the full-fledged mendicant vows of non-posssession (aparigraha), and so could legally possess monasteries and manuscripts. (p. 80)

Last, the topic which most interests me, namely, which manuscripts were actually preserved?

A perusal of the title of the manuscripts indicates that the number of copies of a given manuscript are directly related to its ritual and authoritative roles. We find many copies of texts belonging to the Śvetāmbara ‘canon’, devotional texts used in community rituals, narrative texts used by monks as the bases for sermons, grammars used for the learning of Sanskrit and Prakrit, and texts that are crucial to the mendicant praxis. […] More technical or philosophical works were copied less frequently. (p. 79)

The texts [preserved in Jain libraries, EF] have significantly augmented our understanding of the social, royal, intellectual, and artistic history of Western India. Since the Jains have been quite catholic in their attitudes towards the collection and retention of texts, the bhaṇḍār collections have also included valuable Brāhmaṇical and Buddhist texts that would otherwise have been lost to posterity. (p. 85)

Careful readers will have noted that Cort only speaks of Western India. This is due to the fact that this article is based on his PhD thesis (1989), focusing on North Gujarat. Still, I wonder whether some similar study is available for Jaina libraries in South India…