Do 之 and 的 come from the same word?
的 in its function as a particle is attested in the 四大名著 Four Great Classical Novels, which are written in a vernacular Mandarin-type language, dating from the Ming dynasty. The particle use of 的 is also attested from the Yuan dynasty, when it seems it was adopted for the grammatical particle of the emerging new literary language. Its earliest attestation is in a Song dynasty source (my reference says Coblin, personal communication); another one can be dated to an inscription of 1238 (ABC Etymological Dictionary of Old Chinese, p.208). The consensus is that it is linked to the grammaticalisation of 底 dǐ (root, foundation, bottom), e.g. according to 汉字源 HanZiYuan's entry for 的 (see meaning three). 

The use of 底 for the genitive particle / general nominaliser was a slow process which can be observed through diachronic studies of the extant literature. Note that 的 has functions which can be split into several aspects, and these also emerged gradually.

The lexeme 底 (and its cognates 氐、抵 etc., with Baxter-Sagart Old Chinese reconstruction *tˁijʔ and Baxter Middle Chinese reconstruction *tejX) was semantically extended, and then went through semantic bleaching and subsequent grammaticalisation. This actually covers the whole period from pre-Qin sensu stricto Classical Chinese through the Han to the Tang and Song dynasties. Its grammatical uses came to be very varied by the end of the Tang (including a status suffix for proper nouns, a demonstrative, an intensifier, and even a general interrogative). 

In the southern Song, its use as a genitive became quite well-established [e.g. 療萬病底藥], and then took on functions as a relative clause marker, and then extended to headless clauses [e.g. 碾損老僧腳底], which approaches its role as a nominaliser. It is also during the Tang/Song period that the character 的 emerged in this role as well. The story is told in the 2010 paper "Delexicalizing di: How a Chinese noun has evolved into an attitudinal nominalizer". 

Could 之 then be related to it? The same paper mentions how on phonological grounds that some have postulated that 底 is derived from 之, and others saying it came from 者. But note that there is a fundamental structural difference: Wang Li's work 漢語史稿 states in Chapter 45:


  由此看來，上古漢語這種結構中的動詞（或動詞仂語）近似一種行為名詞 (action noun)，中古以後，在口語中漸漸喪失了這種結構，只有古文作家模仿這種結構寫成書面語言。大約口語重的“的”（“底”）字產生後，這種結構就在口語絕迹了，因為“的”是尾詞（語尾），不是介詞，所以沒有把介詞“之”字這種功能繼承下來。


which, roughly paraphrased, says:


  The verb / verbal phrase in a 之 structure of the Classical age virtually acts as an action noun, but this structure was gradually lost in the oral vernacular of the post-Classical age and was only imitated by scribes in their literary language. After the appearance of 的/底, the structure was lost from the oral language, because 的 is a final particle rather than a preposition, and so could not preserve its structure.


On the other hand, Pulleyblank pulls 者、之 and 底/的 together with the Tibetan demonstrative de, and surmises that the original particle was merely *tV, i.e. the consonant */t/ with any syllabic vowel (Outline of Classical Chinese Grammar, pp.166-7). 之 being *tə, with 者／諸 as *ta, indicates a form of ablaut. It also means that the actual modern pronunciation of 的 is meant to be a relic of the original pronunciation of 之, and that the modern pronunciation of 之 is an evolution of the "reading pronunciation". Hence the ABC Etymological Dictionary of Old Chinese states that 的 is:


  thought to be a col. archaism of the classical 'genitive particle' → zhī1 之


So the consensus seems to agree that it has resisted the sound changes of the ages. Whether there are many other such "colloquial archaisms", that remains to be seen. We'd also have to cover lexemes in other Sinitic varieties too (there's the "obvious" archaism of 筷 vs 箸, found in the Min topolects).