A tweet drew my attention to a monument by a certain Tiberius Claudius Babillus which mentions the library of Alexandria. There is a Wikipedia article about him here, asserting that he died in 59 AD (we will all be wary of anything in this source I am sure). An image is online at Wikimedia Commons here:
The page gives the following source information:
Source: “Forschungen in Ephesos”, Vol. III, Vienna 1923, p.128.
References: IK-17-01, 03042 = AE 1924, 00078 = AE 1927, +00156 = AE 1933, +00251b = AE 1934, +00001
From which I infer that the inscription is from Ephesus. The extremely formulaic nature of Roman inscriptions means that the image fills in some of the missing chunks, where these are routine. There is a clear mention of “(Alexandri)na Bybliothece”
It would be nice to know more about this monument.
From JSTOR I see the text of the inscription on p. 395 (number 78) of R. Cagnat and M. Besnier, “Revue des publications épigraphiques relatives à l’antiquité romaine: 1924,” Revue Archéologique, 5. Série, T. 20 (Juillet-Décembre 1924), pp. 376-421. On Balbillus see p. 143 (number 23) of P.A. Brunt, “The Administrators of Roman Egypt,” Journal of Roman Studies 65 (1975) 124-147, and p. 127 (footnote 60) of Roger Beck, “The Mysteries of Mithras: A New Account of Their Genesis,” Journal of Roman Studies 88 (1998) 115-128.
Thank you very much – useful!