The Norfolk & Norwich Genealogical Society issued a series of "Norfolk Genealogy" volumes. Volume XXI "An Index to Norwich City Officers 1453-1835 edited by Timothy Hawes" in 1989.
On page 13 is the following entry:
"Barrett Christopher, councillor W 1600-01, 1603-15,17-24; auditor 1610-13, 27-33; speaker 1613-15,18; surveyor 1614-15, 32-33; sheriff 1615; coroner 1621-1624; chamberlain's council 1622-23; alderman SS 1625-49; treasurer 1632-33, 43-46; candidate for mayor 1633; mayor 1634; deputy mayor 1647"
Abbreviations:
W = Wymer (a major ward) ; SS = St Stephen (a minor ward).
Edited: I have posted the above details on his profile as a comment.