$p$-adic integrals and Cauchy's theorem
There is an important difference, relevant to the original question, between the two kinds of $p$-adic integrals mentioned by Kevin in his comments. Because I see frequent confusion on this issue, I thought I'd comment.
The 'usual' $p$-adic integrals as you might see in, say, Tate's thesis on L-functions or the adelic theory of automorphic forms, are volume integrals, with respect to a measure, typically on some group. This kind of volume integral can also be easily defined on arbitary varieties, and you will see plenty in Weil's book on Tamagawa numbers, or in papers on motivic integration. Coleman integration, on the other hand, is a $p$-adic analogue of line integrals, and comes up most naturally in discussing the holonomy of vector bundles with connection on a variety over a $p$-adic field (often interpreted as isocrytals). These, therefore, should be the right quantities to relate to a Cauchy formula. However, unfortunately (and fortunately), it doesn't work. The reason is that Coleman integration is a line integral along a canonical path between two points on a variety over the $p$-adics. So there is a canonical holonomy in the theory, at least if you just want to compute it for a bundle with unipotent connection, that is, one that has a strictly upper-triangular connection form. This is where a mysterious 'crystalline' structure on the space of paths is used, whereby there is a unique path invariant under the action of the Frobenius. The notion of a path, by the way, uses the Tannakian formalism in this context. For a very quick overview of this approach, you can look at section 2 of this paper: http://www.ucl.ac.uk/~ucahmki/siegelinv.pdf
Breuil's paper linked from Chandan's answer should provide a more systematic overview.
Anyways, because of the canonical paths in Coleman's theory, there can be no holonomy around a loop, and hence, no Cauchy formula. I was told quite a few years ago by Berkovich that he has a theory of line integrals on Berkovich spaces that are path dependent in interesting ways, but I've never looked into it.
Added: I realize I didn't mention above the connection between holonomy and usual integration of a one-form $A$. You get this by considering the connection $$d+\begin{bmatrix}0& A; \\ 0& 0\end{bmatrix}$$
on the trivial bundle of rank two. One view of Coleman integration is that the holonomy $H_a^b$ from $a$ to $b$ is defined first. And then, the naive integral is defined by the fomula $$H_a^b=\begin{bmatrix}1& \int_a^bA ;\\ 0& 1\end{bmatrix}$$
There exists a theory of the Shnirelman integral providing Cauchy-type formulas for $\mathbb C_p$-valued rigid (Krasner) analytic functions on subsets of $\mathbb C_p$. For a modern exposition see M. M. Vishik, Non-Archimedean spectral theory, J. Soviet Math. 30 (1985), 2513--2554.
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 $\mathbb{C}_p^\times$, 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.