The following does not answer your question, and in fact it is in the negative direction, but it is too long to be added as a comment.
(A): The answer to your question is no, if $V=L,$ as then for any real $r, L[r]\models \Diamond+CH.$
(B): For your stronger question, again we have the following negative consistency result. Assume that $V$ is a model of $ZFC$ which satisfies the following (such a model exists):
Every tree of height and size $\omega_1$ is special,
$2^{\aleph_0}=\aleph_2.$
The by a theorem of Todorcevic (see "some combinatorial properties of trees"), any forcing notion which adds a new subset of $\omega_1,$ collapses $\aleph_1$ or $\aleph_2.$ It means that in all generic extensions $V[r]$ of $V$, where $r$ is a new real, the power function changes, and cardinals are collapsed.
These results show that, we need some preparation model to construct your required model $V$.