Berlin has a few.
Most famous is Fischerinsel, named after an old fisherman's settlement that datedback to the 13th century. The northern end of the island is home to some of Berlin's museums, and therefore called Museum Island. The southern end--still called Fischerinsel--somehow survived WWII as an intact pre-industrial district, only to fall to the commies and their Platenbau in the '60s.
https://www.google.com/maps/@52.5152.../data=!3m1!1e3
Alt Spandau is another: a medieval old city at the confluence of the Spree and Havel, an old miller's canal or something separates it from land.
https://www.google.com/maps/@52.5374.../data=!3m1!1e3
On the opposite side of the city, Alt Köpenick sits at the confluence of the Spree and Dahme rivers, and has its own moat to complete its insulation.
https://www.google.com/maps/@52.4458.../data=!3m1!1e3
I guess you could argue that entire districts like Moabit and North Charlottenburg are islands, since they're surrounded by river and canal, but they don't feel like it.
There's also Eiswerder, just north of Alt Spandau, but I've never been there and it doesn't look super exciting.
https://www.google.com/maps/@52.5508.../data=!3m1!1e3