This article is from the California Driving (and Surviving) FAQ, by "George J Wu" firstname.lastname@example.org with numerous contributions by others.
Not if you don't break any laws. If you do break the law and don't want to
sit in jail, it helps to have positive identification: "Whenever any person
is arrested by a peace officer for a misdemeanor, that person shall be
released [...] unless [...] [t]he person could not provide satisfactory
evidence of personal identification" [Penal Code 853.6(i)(5)]. Note that
most traffic tickets are not for misdemeanors but infractions, and "all
provisions of law relative to misdemeanors shall apply to infractions"
[Penal Code 19d].