We can define padic Bernoulli polynomials by using qintegral on $\mathbb{Z}_p$ and Taekyun Kim's method.
But how can we define padic polyBernoulli numbers and polynomials by using integral on $\mathbb{Z}_p$?
We can define padic Bernoulli polynomials by using qintegral on $\mathbb{Z}_p$ and Taekyun Kim's method. But how can we define padic polyBernoulli numbers and polynomials by using integral on $\mathbb{Z}_p$? 


I think what you are looking for is: $$B_n^{k}(x)=\frac{Li_k (1e^{t})}{t}\int_{\mathbb{Z}_p}(x+y)^ndy$$ for $n\geq0$, and $Li_k$ the polylogarithmic function. See for example, "PolyBernoulli Polynomials and Their Applications", by Kim, Jang and Seo. 

