The divergence of a vector field is a measure of how much the vector field spreads out or converges at a point. It is defined as follows:
$$\nabla \cdot \overrightarrow{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$$
In this case, the vector field is $\overrightarrow{A} = x\hat{\imath} + y\hat{\jmath} + z\hat{k}$. The divergence of this vector field is therefore:
$$\nabla \cdot \overrightarrow{A} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1$$
Therefore, the correct answer is $\boxed{1}$.
The other options are incorrect because they do not represent the divergence of the vector field $\overrightarrow{A}$.