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No, not every new real in the Silver extension is generic for Silver forcing. To see this, take any new real $x$ in the extension, and form a new real $y$ by doubling every digit of $x$. So if $x=011010\ldots$, then $y=001111001100\ldots$. The real $y$ cannot be Silver generic, since it is dense in Silver forcing to violate the digits-doubling property. Given any condition, look at the first bit not specified, and then fill that bit (and possibly the next) in such a way that the condition is violated.