Yup! Bananas are grown in the warm, tropical corners of the US, namely Hawaii and Florida. People also grow bananas in other warm, non-tropical areas like California, Louisiana, Arizona, Texas. And there are folks who try to grow bananas in places like New Jersey and Connecticut. I find that a bit like forcing a square peg into a round hole, but there are some cold-hardy varieties out there that would survive, and maybe even produce fruit if grown in a greenhouse.

Keep in mind that instances of banana growing in the US are either for a small local business, research purposes, or home gardeners. They're not grown in huge quantities because our climates aren't ideal. Bananas need consistently warm temperatures. If it's too hot or too cold, the plant will stop/slow growth. The majority of the bananas we eat are grown in Latin America and the Caribbean area.

California: http://webebananas.com/
Florida: Welcome to Going Bananas of Homestead, Florida!
Arizona: Growing Bananas In Phoenix Arizona
Louisiana: Bananas , Banana sets fruit after more than 30 years

In Hawaii, they're all over the place. When I was on the Big Island, I saw them in people's backyards, on plantations. Grocery stores had half Hawaiian-grown bananas, half imported. They grow a lot of apple bananas, a variety more tart than the regular Cavendish variety: Home of the Hawaiian Candy Apple Banana

Here's a photo of a banana tree I saw on the Big Island:

Il frutto viene poi trasportato nei porti per essere imballato in navi refrigerate chiamate reefers (le banane impiegano dai sei ai dodici giorni per arrivare negli Stati Uniti). ... Quando le banane arrivano al porto di destinazione, vengono prima inviate in camere di maturazione (un processo che coinvolge il gas etilene) e poi spedite ai negozi.