Ykhjuy

In many languages the word for date (a specific day, such as January 2, 2019) seems to come from the Latin participle datus: we have the English "date", the Italian "data", the Swedish "datum", and others.
Do these words come from a Latin phrase such as dies datus/data?
If yes, when is it attested?
Latin origin seems evident, but I wonder whether it is classical or a later development.

For reference, this is what the OED has to say:

Etymology: < Anglo-Norman dat, Anglo-Norman and Middle French date
(Middle French datte ; French date ) regnal year (1230 or earlier),
date (specified on a document) (1281 in Old French), date (more
generally) (1314 or earlier) < post-classical Latin data (6th cent.;
frequently from 11th cent. in British sources), use as noun (see note)
of feminine singular of datus , past participle of dare to give (see
datum n.). In classical Latin, the date of a letter was expressed by a
phrase such as data xiiii K. Maias de Tarentino ‘(letter) sent from
Tarentum on 18th April’ (Cicero Letters to Atticus 3. 6. 1), litteras
datas a litoribus Britanniae proximis a. d. vi Kal. Octobr. ‘letter
sent from the nearest shores of Britain on 26th September’ (Cicero
Letters to Atticus 4. 18 . 5), litterarum datarum dies prid. Kal. Ian.
‘the day of a letter sent on 31st December’ (Cicero Letters to Atticus
6. 1. 2). Hence data , the first word of the formula, came to be used as a term for the time and place stated therein. Compare Catalan data
(14th cent.), Spanish †data (mid 13th cent.), Portuguese data (13th
cent.), Italian data (a1556). Compare also (after similar dating
formulae in post-classical Latin using the neuter singular datum (13th
cent.)) Middle Low German dātum , dāte , Middle Dutch, Dutch datum ,
Middle High German dātum (German Datum , in early modern German also
date ).

There seems no reason to doubt that the English 'date' and similar words in other languages derive from the perfect participle of dare.

A very plausible origin is as a a relic of litterae datae which was used to indicate the date on which a letter was handed over to a courier. This is also shown in the singular form [epistola] data quoted in the extract from the OED provided by @fdb.