I always laugh when I hear the Beatles referred to as "classic rock." Yes, the played some rock. And folk, and classical, and country, and pop, and Tin Pan Alley, and swing. Hey, the even played an Italian-style Tarantella ("Yellow Submarine")!
They also played music that was heavily influenced by South American dance music such as rhumba and Bossa Nova.
Historically, it makes sense. Bossa Nova took the world by storm in the late 1950s - early 1960s, which is exactly when the Beatles began playing together. They arranged and recorded the Broadway show tune "Till There Was You" (from The Music Man) with a light Bossa Nova-style beat, slightly modified for the pop market.
Their early classic "And I Love Her" is another one like this. The fun thing for us pianists is that we can use the Beatles original concept as a jumping off point to either go deeper into the Bossa style, which is closely related to jazz, or away, towards folk music for instance.
Here's a resource I put together to give you an overview on "And I Love Her." You'll start by learning more about the song itself, and then you'll get some ideas on how to improvise your own arrangement by developing the Bossa Nova aspect more fully.
Like Irving Berlin, George Gershwin and Duke Ellington before them, the Beatles wrote such perfect songs that we can take them anywhere we wish. Just like the Beatles themselves did. Good luck and have fun!