<p>This paper extends the Rubio de Francia extrapolation theory to the ball Banach function spaces on the one-sided setting. The extrapolation theory obtained in this paper relies on the boundedness of the one-sided maximal operators instead of the Hardy-Littlewood maximal function. Our main result yields the boundedness of the one-sided singular integral operators and the one-sided martingale transforms on the ball Banach function spaces. We also obtain the Coifman inequalities for the one-sided singular integral operators on the ball Banach function spaces. A concrete example of ball Banach function space for our main results is given, namely, the one-sided Herz spaces with variable exponents. The Hardy-Littlewood maximal function is not bounded on the one-sided Herz spaces with variable exponents while the one-sided maximal operator is bounded. It shows that the extrapolation theory relied on the boundedness of the Hardy-Littlewood maximal function cannot be applied to the one-sided Herz spaces with variable exponents and the results from our extrapolation theory cannot be covered by the extrapolation theory relied on the boundedness of the Hardy-Littlewood maximal function.</p>