A number of commenters mentioned New York City, which does have several Really Narrow Streets south of Fourteenth Street. With the help of this Forgotten NY post, I've identified the narrowest streets in Manhattan and got a rough measure of their widths using Google Earth. There are ten of them under twenty feet wide:
View New York's Really Narrow Streets in a larger map
Interestingly, many of them are what I'd call alleys rather than Really Narrow Streets. What's the difference between an alley and a Really Narrow Street? I'd say it's one of function. Really Narrow Streets are the main pedestrian access to shops and residences. Alleys are used to allow cars and trucks to access rear entrances to loading docks and parking lots.
On Tuesday I happened to be walking through Lower Manhattan and I snapped my own cell phone pictures of some of these Really Narrow Streets/alleys.
Note that Liberty Place is covered with ugly scaffolding. So is Catherine Lane:
It seems to be almost entirely used as a driveway for that parking garage.
Cortlandt Alley is definitely used as an alley for motor vehicle access. Nate Berg has a great interview with Nick Carr of the Scouting Report about Cortlandt Alley's film career:
The big thing I always get asked to find are dank dilapidated alleys, and New York City has, like, 5 alleys that look like that. Maybe four. You can’t film in three of them. So what it comes down to is there’s one alley left in New York, Cortlandt Alley, that everybody films in because it’s the last place. I try to stress to these directors in a polite way that New York is not a city of alleys.
It seems like it would be possible, maybe with some zoning changes, for some foresighted developer to take one of these streets, open a bunch of vibrant storefronts and market the hell out of it so that every tourist stops there for shopping and street food after they visit Ground Zero. It might take a year or two, but I could see one of them becoming New York's answer to Bearskin Neck or the Rue Sous le Cap. In Strong Towns terms, think how much tax revenue the city would get from such a tiny amount of infrastructure.