No dictionary I've checked lists such a usage. The closest to what I'm looking for is DMLBS listing a meaning "2.b (w. ellipsis of lingua)", but the two examples it gives display ordinary (in effect mandatory) syntactic ellipsis to avoid repetition - the adjective is not substantive, but unambiguously headed. It also gives 2.c "(w. ellipsis of litterae)", with the beautiful example "solempnem in ~is fuit sermonem" (unfortunate that I can't find the source). Thus it doesn't seem like they've been able to find an example even in Medieval British Latin. Though, apparently, Old Norse has a feminine latīna referring to the Latin language: "W.[=Weiterleben].: an.[=altnordisch] latīna, F., Latein, lateinische Sprache", and in Celtic languages it's also feminine - but so appear to be all other adjectival language names in Irish Gaelic, Welsh and Manx.
The three things that come to my mind when I see a standalone Latīna in a sentence are: 1. a Latin woman; 2. via Latīna; 3. porta Latīna. And as a bonus, 4. the modern Italian city of Latina. As a Neut.Pl., it would mean "Latin things". Gaffiot lists a contextual meaning "pl. Latina Cic. Arch. 23, les œuvres en latin", which to me seems like a simple switch-up of the written versūs and the unwritten carmina (that's what my brain told me, and it took me several reads to make sure there was in fact no carmina in that sentence).
Thus, to the best of my knowledge, Latin simply doesn't use Fem.Sg. or Neut.Pl. substantive adjectives as language names. When there's a need of conveying this with a single adjective, Neut.Sg. adjectives are found instead: vertere ex Graecō in Latīnum. Granted, I've yet to see an example of one used as a subject (*Latīnum mē dēlectat), but this seems accidental - and there are examples of non-prepositional use in DMLBS. Besides, the same form is found in the expression Latīnum esse "to be a Latin word or turn of phrase; to be Latin". This form is what looks to be continued in all the modern Romance languages without syntactic limitations (Italiano, Español, Français, Sardu) apart from Daco-Romanian (Română) and perhaps any related dialects.
I myself would like to find an example of this usage in anything that can reasonably be called fluent Latin, even if distinctly... medieval, this side of the 20th century. So far I haven't been able to, which makes me conclude that this usage is not only un-Ciceronian, or by chance hasn't found its way into the dictionaries, but is simply not Latin. In cases like this the reasonable thing to do is to simply accept the vast majority of speakers' opinion on how something is and isn't said in the language you're learning, even if the reasons why might not be readily explicable.
I don't believe that the unheaded usage "with lingua implied" is grammatical, nor that the use of standalone feminine adjectives to refer to language names is lexicalised in any sort of Latin. I'm all but certain that it's confined to the Latin of 20/21st century English speakers, in particular those who have been exposed to "Wheelock's Latin" where it's found in a sentence as hair-raising as the rest of that book - or to too much Vicipaedia.