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 $Z_p$ we denote the ring of p-adic rational
integers, Q denotes the ﬁeld of rational numbers, $Q_p$ denotes the ﬁeld of p-adic
rational numbers, and $C_p$ denotes the completion of algebraic closure of $Q_p$. Let
N be the set of natural numbers and $N^∗ = N ∪ { 0 } $. The p-adic absolute value is
deﬁned by $|p|_p =\frac{1}{p}$. We assume $|q − 1|_p < 1 $ as an indeterminate. Let
$UD(Z_p)$ be the space of uniformly diﬀerentiable functions on $Z_p$. For a positive
integer $d$ with $(d, p) = 1$, set

$$X = X_d = lim_{←n} Z/dp^nZ,$$

$$X^∗ = ∪_{0< a< dp , (a,p)=1} a + dpZ_p$$

and
$a + dp^nZ_p = \{x ∈ X | x ≡ a (mod dp^n)\}$.

where $a ∈ Z$ satisﬁes the condition $0 ≤ a < dp^n$.

Firstly, for introducing fermionic p-adic q-integral, we need some basic information which we state here. A measure on $Z_p$ with values in a p-adic Banach space

B is a continuous linear map
$$f →\int f(x)\mu =\int_{Z_p} f(x)\mu(x)$$

from $C^0(Z_p,C_p)$, (continuous function on $Z_p$ ) to $B$. We know that the set of
locally constant functions from $Z_p$ to $Q_p$ is dense in $C^0(Z_p,C_p)$ so explicitly, for all $f ∈ C^0(Z_p,C_p)$, the locally constant functions
$$f_n =\sum_{i=0}^{p^n−1} f(i)1_{i+p^nZ_p} → f$$ in $C^0$.

Now, set $\mu (i + p^nZ_p) = \int_{Z_p}1_{i+p^nZ_p}\mu$, then $\int _{Z_p}f\mu$ is given by the following Riemannian sum,

$$\int_{Z_p}f\mu = lim_{n→∞}\sum_{i=0}^{p^n−1}f(i)\mu(i + p^nZ_p)$$.

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