Rockridge is definitely part of Oakland, but Piedmont is a separate city. Piedmont seceded from Oakland in 1907 and is completely surrounded by Oakland. I've never heard of any other city engulfed within another.
This property of Piedmont is not even unique within the Bay Area! Fremont (the fourth-largest city in the Bay Area) has an enclave of its own, Newark, which is its own city yet completely surrounded by Fremont.
Just for fun.. The Vatican City is surrounded by Rome, though it's not only a city but a nation state. If you want California, though, Los Angeles has several enclaved cities, such as San Fernando.