Is it possible for an object to simultaneously have a speed of zero but an acceleration that is not zero?
Take something with a bit of weight to it (perhaps a coin) and hold it with the tips of your fingers. Count down from three (3... 2.... 1...), let go, and then catch it again with those same fingertips. If you are fast, you'll catch it with the coin not having moved at all.
At the instant you let go of the coin, its speed is zero. But it is experiencing an acceleration, that of gravity. That coin then is experiencing acceleration that is not zero but its speed is zero.
Let's take that idea one step further. Everything on the surface of the Earth is experiencing acceleration due to gravity but much of it isn't in motion. A rock sitting on the ground, whether on a mountaintop, on a street, or on the seafloor.
And in fact, when you held up that coin, you too were experiencing acceleration (and if you weren't moving, no speed). And you still are.