The answer to the short version of your question is: yes, there is a $p$-adic theory of integration.
As to whether an analogue of Cauchy's theorem holds in such a theory, I assume you are thinking of Cauchy's integral formula, and I work with the theory of Coleman. That formula amounts to the statement that the complex logarithm is a multivalued function. But over the $p$-adics, for each choice of $\mathscr{L}\in\mathbb{C}_p$ there is a so-called branch of the $p$-adic logarithm, which is defined on all $\mathbb{C}_p$, \mathbb{C}_p^\times$, uniquely determined by the fact that it behaves as expected for the product and it extends the convergent power series $$\log(1+x)=\sum_{n=1}^\infty\frac{(-x)^n}{n}$$ of$x\in\mathbb{C}_p$with$v_p(x)>0$. Each of these provides a primitive of the function$f(x)=\frac{1}{x}$on$\mathbb{C}_p$, and the Fundamental Theorem of Calculus´´ satisfied by Coleman's theory of$p$-adic line integrals, will tell you that the integral of$f(x)$along a closed loop around$0\in\mathbb{C}_p$is zero. A good reference: MR0782557 (86j:14014) Coleman, Robert F. Torsion points on curves and$p$-adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168. 1 The answer to the short version of your question is: yes, there is a$p$-adic theory of integration. As to whether an analogue of Cauchy's theorem holds in such a theory, I assume you are thinking of Cauchy's integral formula, and work with the theory of Coleman. That formula amounts to the statement that the complex logarithm is a multivalued function. But over the$p$-adics, for each choice of$\mathscr{L}\in\mathbb{C}_p$there is a so-called branch of the$p$-adic logarithm, which is defined on all$\mathbb{C}_p$, uniquely determined by the fact that it behaves as expected for the product and it extends the convergent power series $$\log(1+x)=\sum_{n=1}^\infty\frac{(-x)^n}{n}$$ of$x\in\mathbb{C}_p$with$v_p(x)>0$. Each of these provides a primitive of the function$f(x)=\frac{1}{x}$on$\mathbb{C}_p$, and the Fundamental Theorem of Calculus´´ satisfied by Coleman's theory of$p$-adic line integrals, will tell you that the integral of$f(x)$along a closed loop around$0\in\mathbb{C}_p$is zero. A good reference: MR0782557 (86j:14014) Coleman, Robert F. Torsion points on curves and$p\$-adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168.