Heidegger's Bible Handbook: 1 Peter: Time of Writing
7. The opinion of Baronius concerning the time of writing. That it appears to have been written at Antioch in 52 AD, and delivered by Sylvanus.
Concerning the time of writing also, we are able to pronounce nothing with certainty. If we believe Baronius, it was written in 45 AD, and so was extant as the first of all the New Testament writings. Indeed, that out of 1 Peter 4:16, where the nomenclature of Χριστιανοῦ/Christian is found, demonstrates that it was written after the time in which the disciples of Christ began to be called Χριστιανοὶ/Christians at Antioch. Which Baronius refers to 43 AD. Out of Acts 15:35 it is evident that Paul and Barnabas, with many others, taught and preached the Gospel at Antioch. That Peter came to the same place, with those being present, is proven out of Galatians 2:11, 13. From Antioch, Paul departed with Silas, with Barnabas and Mark left behind, for the Asiatic Churches, Acts 15:39. That happened around 52 AD. And it is likely that at that same time Peter wrote this Epistle, and entrusted it to Sylvanus to be exhibited to the Jews dispersed everywhere.
 Acts 11:26.