California police rarely have the power to arrest you for not producing official identification. However, the situations in which they could detain you are common enough that you might benefit from knowing your rights.
First and foremost, know that you could face penalties if you fail to show your driver's license or identify yourself during a traffic stop. This discussion is focused more on the potential consequences of withholding identification information in other situations.
California does not have a so-called "Stop and Identify" statute. The presence of this law would allow police officers to arrest you if you did not give them ID. However, even though you would not have to identify yourself to an officer should he or she ask, you could still face detainment and arrest in certain situations by failing to comply.
As discussed in an LA Times article about a high-profile detainment, police might be able to arrest you if you fail to identify yourself during a misdemeanor citation. This is notable because most situations involving misdemeanors would not end in you being thrown in jail.
There are many other factors that might affect a law enforcement professional's decision to detain you. For example, ICE agents could behave differently than state patrol officers. Another example: if you are part of an active investigation, police might be able to hold you on suspicion of a crime — suspicion you might be able to clear up by providing ID.
You should probably assess each police encounter you have on an individual basis: almost anything you do or say could become material in a prosecutor's case. Please remember that it is a policeman's job to enforce the law, prevent crimes and — most relevant to this discussion — to collect information that could lead to fines, penalties or even your incarceration. Therefore, please do not regard this as specific legal advice. It is only general information.