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In simple models, yes. They do.

In a more correct and more complex model, it is not possible to determine the exact position of an electron at any point, (as they are quantum) but areas of high probability for the electrons to be in exist, they are called orbitals, and yes, they surround the nucleus in a manner of speaking.

In metallic bonding, the electrons de-localise from the orbitals, and they float around the substance. These electrons are not surrounding the nucleus.

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