  We establish new upper bounds for Berezin number and Berezin norm of operator
matrices, which are refinements of the existing bounds. Among other bounds, we
prove that if $A=[A_{ij}]$ is an $n\times n$ operator matrix with
$A_{ij}\in\mathbb{B}(\mathcal{H})$ for $i,j=1,2\dots n$, then
  $\|A\|_{ber} \leq \left\|\left[\|A_{ij}\|_{ber}\right]\right\|$ and
$\textbf{ber}(A) \leq w([a_{ij}]),$ where $a_{ii}=\textbf{ber}(A_{ii}),$
$a_{ij}=\big\||A_{ij}|+|A^*_{ji}|\big\|^{\frac{1}{2}}_{ber}
\big\||A_{ji}|+|A^*_{ij}|\big\|^{\frac{1}{2}}_{ber}$ if $i<j$ and $a_{ij}=0$ if
$i>j$. Further, we give some examples for the Berezin number and Berezin norm
estimation of operator matrices on the Hardy-Hilbert space.
