In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M3"> <a:mi>p</a:mi> </a:math> -adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M4"> <c:mi>p</c:mi> </c:math> -adic variable exponent Lebesgue spaces.