Yes — very slightly, and it is completely fine. Bananas contain potassium, a tiny fraction of which is the naturally radioactive isotope potassium-40, so eating one banana exposes you to roughly 0.1 microsievert of radiation. Your body regulates its potassium tightly, so that potassium does not accumulate, and bananas remain perfectly safe to eat.
Potassium is an essential mineral, and bananas are famously rich in it. Natural potassium is almost entirely stable, but a very small fraction of it — about one atom in every ten thousand or so — is potassium-40, an isotope that has existed since before the Earth formed and decays extremely slowly. Wherever potassium goes, a trace of that gentle radioactivity goes with it: bananas, potatoes, beans, and your own body, which carries potassium in every cell.
That last part is the perspective check. You are slightly radioactive yourself, for exactly the same reason the banana is. Radioactivity at these trace levels is a normal feature of the natural world, not a contamination event.
Because potassium is on a budget your body enforces. Your system holds its total potassium within a tight range, taking in what food provides and sending the excess out. Eat more potassium and you simply excrete more — the internal total, and with it the internal dose, stays about the same. That is why the banana comparison is used in classrooms in the first place: it shows how small everyday doses are, and why the dose from a banana on Monday does not stack onto the dose from a banana on Tuesday.
For scale, ordinary background radiation from the ground, the air, and space delivers far more than a banana does, every single day, whether you eat fruit or not. The banana is a rounding error in a budget you were already living with.
See the short version: "Your Banana Is Radioactive (And That's Fine) — Things You Didn't Know" on LOE PRESSURE (Science #35).
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Sources: Encyclopaedia Britannica, "Are Bananas Radioactive?"; BBC Science Focus coverage of potassium-40 and the banana equivalent dose.