For a variety $X$ defined over $\mathbb{Q}$, there's a (functorial) comparison isomorphism $$ H^i_{dR}(X)\otimes\mathbb{C}\to H^i_B(X,\mathbb{Q})\otimes\mathbb{C}. $$ If we pick $\mathbb{Q}$-bases for $H^i_{dR}(X)$ and $H^i_B(X,\mathbb{Q})$, the matrix entries of the comparison are complex numbers called periods of $X$, or more specifically periods of the motive $h^i(X)$.

Is there a comparison of cohomology theories that leads to a definition of 'period' for $p$-adic numbers, so that some but not all elements of $\mathbb{Q}_p$ are periods?

I have a specific example in mind: Riemann zeta values $\zeta(k)$, $k\geq 2$ an integer, are periods. An analogue in $\mathbb{Q}_p$ are the values $\zeta_p(k):=L_p(k,\omega_p^{1-k})$ of the $p$-adic $L$-function of Kubota-Leopoldt, where $\omega_p$ is the Teichmuller character. I'm hoping there's a sense in which $\zeta_p(k)$ is a period.

For $X$ a sufficiently nice variety over $\mathbb{Q}_p$, if I understand correctly there's a comparison isomorphism $$ H^i_{dR}(X)\otimes B \to H^i_{et}(X,\mathbb{Q}_p)\otimes B, $$ where $B$ is Fontaine's ring of $p$-adic periods. Elements of $B$ which show up as matrix coefficients with respect to $\mathbb{Q}_p$ bases of $H^i_{dR}(X)$ and $H^i_{et}(X,\mathbb{Q}_p)$ should also be called periods. Since both cohomologies in this comparison started out defined over $\mathbb{Q}_p$, every element of $\mathbb{Q}_p$ is a period in this sense.