  Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$
elements. This paper focuses on the study of values of a generic family of
hypergeometric functions in the $p$-adic setting which we denote by
${_{3n-1}G_{3n-1}}(p, t),$ where $n\geq1$ and $t\in\mathbb{F}_p$. These values
are expressed in terms of numbers of zeros of certain polynomials over
$\mathbb{F}_p$. These results lead to certain $p$-adic analogues of classical
hypergeometric identities. Namely, we obtain $p$-adic analogues of particular
cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the
zeros of these functions. For instance, if $n$ is odd then we obtain zeros of
${_{3n-1}G_{3n-1}}(p, t)=0$ under certain condition on $t$. In contrast we show
that if $n$ is even then the function ${_{3n-1}G_{3n-1}}(p, t)$ has no
non-trivial zeros for any prime $p$.
