  We bound double sums of Kloosterman sums over a finite field ${\mathbb
F}_{q}$, with one or both parameters ranging over an affine space over its
prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite
fields analogues of a series of recent results by various authors in finite
fields and residue rings. Our results are based on recent advances in additive
combinatorics in arbitrary finite field.
