Can an officer arrest a person for DUI in a private parking lot?

Most DUI statutes say driving on the highways or (areas open to the public). This is an area open to the public.

Driving includes driving on private property.

In Washington, it is not a defense that the driving was in a private parking lot. Now if the car was safely parked, that is a different issue and a potential defense.

Yes the cop can. You need to hire a DUI specialist, and do it soon because you have only 10 days to save your license.

As noted in a dissenting opinion by Appellate Judge Wise in 2008 (Hinson v. State), there is no real case law that supports the stance that you can not get a DUI on private property. However, in Lunceford v. City of Northport, a 1988 case, a DUI conviction was upheld when a driver was found intoxicated in his vehicle that was in a store parking lot. Even if Alabama law could be construed so as to allow for intoxicated driving on private property, Judge Wise points out that there is a distinct difference between operating a vehicle under the influence while on private property that you own and operating a vehicle under the influence while on private property that you do not own. This is especially the case when you are dealing with a store's parking lot that can only be entered or exited by driving on public roadways. Lastly, although it is technically a private parking lot, you are in a public area in which other people and their property are present. Laws that prohibit driving under the influence were created with the intent of protecting people's property and well-being. I do not foresee any court dismissing your case simply because you were operating your vehicle under the influence in a store parking lot as opposed to a roadway. This would undermine the congressional intent for enacting laws against driving under the influence.

In California, yes they can.