In recent decade, several number theorists for instance, professor T. Kim, H. Srivastava, Serkan Araci, extended $q$-integral concept for $p$-adic numbers and found several combinatorial identities for Bernoulli, Euler and Genocchi numbers by using this new method.
I try to briefly explain this method here.

Assume that $p$ be a ﬁxed odd prime number. By $\mathbb Z_p$ we denote the ring of $p$-adic rational integers, $\mathbb Q$ denotes the ﬁeld of rational numbers, $\mathbb Q_p$ denotes the ﬁeld of $p$-adic rational numbers, and $\mathbb C_p$ denotes the completion of algebraic closure of $\mathbb Q_p$.
The $p$-adic absolute value is deﬁned by $|p|_p = 1/p$. We assume $|q − 1|_p < 1 $ as an indeterminate. Let $UD(\mathbb Z_p)$ be the space of uniformly diﬀerentiable functions on $\mathbb Z_p$. For a positive integer $d$ with $(d, p) = 1$, set

$$X = X_d = \varprojlim_n \mathbb Z/dp^n \mathbb Z,$$

$$X^∗ = \bigcup_{0< a< dp}^{(a,p)=1} a + dp \mathbb Z_p$$

and $a + dp^n \mathbb Z_p = \{x \in X \mid x \equiv a \mod {dp^n}\}$, where $a \in \mathbb Z$ satisﬁes the condition $0 \le a < dp^n$.

Firstly, for introducing fermionic $p$-adic $q$-integral, we need some basic information which we state here. A measure on $\mathbb Z_p$ with values in a $p$-adic Banach spaceB is a continuous linear map $$f \mapsto \int f(x)\mu = \int_{\mathbb Z_p} f(x)\mu(x)$$

from $\mathcal C^0(\mathbb Z_p,\mathbb C_p)$, (continuous function on $\mathbb Z_p$) to $B$. We know that the set of locally constant functions from $\mathbb Z_p$ to $\mathbb Q_p$ is dense in $\mathcal C^0(\mathbb Z_p, \mathbb C_p)$ so explicitly, for all $f \in \mathcal C^0(\mathbb Z_p, \mathbb C_p)$, the locally constant functions
$$f_n =\sum_{i=0}^{p^n−1} f(i)1_{i+p^n \mathbb Z_p} \to f$$ in $\mathcal C^0$.
Now, set $\mu (i + p^n \mathbb Z_p) = \int_{\mathbb Z_p}1_{i+p^n\mathbb Z_p}\mu$, then $\int_{\mathbb Z_p}f\mu$ is given by the following Riemannian sum,

$$\int_{\mathbb Z_p}f\mu = \lim_{n \to \infty} \sum_{i=0}^{p^n−1}f(i)\mu(i + p^n \mathbb Z_p)$$.

T. Kim introduced $\mu$ as follows: $\mu_{−q}(a + p^n \mathbb Z_p) ={(-q)^a} / {[p^n]_{-q}}$ for $f \in UD(\mathbb Z_p)$, which is famous to the fermionic $p$-adic $q$-integral on $\mathbb Z_p$ and you can find the applications of this definition in several papers which about $q$-Bernoulli numbers and polynomials. See here.