Recursive Algorithm Copying Array vs. Time Complexity
The first point is that as in the most binary search implementations we are passing an array with their sub-indices each time, and in the common programming languages, arrays are passed by reference this problem cannot happen.

However, if this copy happened each time this would be $n+\frac{n}{2} + \frac{n}{2^2} + \cdots + \frac{n}{2^{\log(n)}} = \Theta(n)$.
Hence, time complexity of this algorithm which copies half of the passed array would be $\Theta(n)$.

Moreover, if the implementation passed the whole of the array each time and control the search using sub-indices, the time complexity would be worse and it would be $\Theta(n\log(n))$, as each time ($\log(n)$) copy the whole of the array with size $n$.