What is the value of $p$-adic $\zeta$-function at positive integer point?

If you use this definition, then $\zeta_p(k)$ is zero at negative even integers $k$, so by a $p$-adic continuity argument, it must also be zero at positive even integers.

What about the odd integers? At $k = 1$ there is a pole, unsurprisingly. At odd $k \ge 3$ the value is extremely mysterious, just as the complex zeta values $\zeta(k)$ are. There is an interpretation of the odd $p$-adic zeta values in terms of a $p$-adic regulator map in $K$-theory (see this question), but this is tough to get explicit information out of.

As an example of how little we understand these numbers, I believe it's an open problem whether the values $\zeta_p(k)$ for odd $k \ge 3$ are always non-zero, although this is certainly expected.

I agree nothing much is known, but there are a number of formulas linked to the values at positive integers of p-adic L-functions: see Section 11.3 of my GTM 240 book (sorry for the self-advertisement).

Let me mention some irrationality results about p-adic zeta values:

In 2005, Frank Calegari arxiv link proved that for $p=2,3$, the p-adic zeta values $\zeta_p(3)$ is irrational (hence nonzero). It can be viewed as the p-adic analogue of Apéry's theorem. Later, Frits Beukers arxiv link gave an alternating proof of Calegari's results (and the irrationality for some other p-adic L-values).

In 2010, Pierre Bel arxiv link obtained some partial results on the irrationality of p-adic Hurwitz zeta values. See also his 2018 paper on $\zeta_p(4,x)$.

Very recently (in 2020), Johannes Sprang arxiv link showed that $$ \dim_{K}(K+\zeta_p(2)K+\zeta_p(3)K+\cdots+\zeta_p(s)K) \geqslant \frac{(1-o(1))\log s}{2[K:\mathbb{Q}](1+\log 2)}, $$ which is the analogue of Ball-Rivoal theorem for p-adic zeta values.