Yes. And no. And yes.
There are an infinite number of frequencies between any two notes, but Western music is based on the piano keys, and to it anything in between is not considered a note. There are other musical traditions that use more notes, like the Arab Tone System that includes the frequencies halfway between the piano keys.
Also, since the precise pitch is arbitrary, you could tune a piano to have all of its notes halfway between the standard ones, and it would sound perfectly fine to most people.
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