p-adic poly-Bernoulli numbers We can define p-adic Bernoulli polynomials by using q-integral on $\mathbb{Z}_p$ and Taekyun Kim's method.
But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $\mathbb{Z}_p$? 
 A: I think what you are looking for is:
$$B_n^{k}(x)=\frac{Li_k (1-e^{-t})}{t}\int_{\mathbb{Z}_p}(x+y)^ndy$$
for $n\geq0$, and $Li_k$ the polylogarithmic function.
See for example, "Poly-Bernoulli Polynomials and Their Applications", by Kim, Jang and Seo.
A: There is no thing as "p-adic Bernoulli polynomials". One can give an equivalent definition of the (usual) Bernoulli polynomials using $p$-adic methods, explicitly, the Volkenborn integral. This definition is certainly not a "Taekyun Kim's method", and was first treated by Arnt Volkenborn in his thesis, inspired by some $p$-adic sums used by Kubota and Leopoldt in their classic article about the $p$-adic zeta function. (See Volkenborn papers and, of course, Kubota and Leopoldt's article). 
Multiple Bernoulli polynomials are easily defined by using multiple Volkenborn integration. A recent and nice reference is Tangedal and Young's "On p-adic multiple zeta and log gamma functions", freely available in the arxiv. Another reference is Kashio's paper on Shintani's methods, in which he implicitly used Volkenborn integration
Now, poly-Bernoulli numbers and polynomials, are treated from a $p$-adic point of view in Paul Thomas Young's papers on $p$-adic Arakawa-Kaneko zeta functions, which is a natural treatment of this polynomials. Probably, a $p$-adic (ad-hoc) definition could be given in terms of Coleman's $p$-adic polylogarithm and Volkenborn integration, in analogy to the complex case.
For an infinite class of poly-multiple-twisted-etc-Bernoulli polynomials, then you can take a look at Kim et al's papers. Caution: q-Volkenborn integration is not due to Kim, and it was perhaps first treated by Junya Satoh (JOURNAL OF NUMBER THEORY 31, 346-362 (1989)).
