If we split between second-generation (i.e, both parents born in Germany) as "native" and everyone else (i.e, anyone with at least one parent not born in Germany) as "immigrants", I'd guess that number would skew heavily towards "immigrants" in all major cities. I could be wrong, but I haven't found a way to reliably check the data.
Doubtful, honestly, though from what you described with Berlin, this feels like more of a problem of Berlin being a very transient city. A lot of people never fully settling there, especially so called expats, despite living there for decades. A lot of people being part of the cosmopolity, moving from one hype city, one startup, to the other. That’s okay, but it doesn’t work for everyone.