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):

1) Every tree of height and size $\omega_1$ is special,

2) $2^{\aleph_0}=\aleph_2.$

The by a theorem of Todorcevic (see "[some combinatorial properties of trees](http://blms.oxfordjournals.org/content/14/3/213.short)"), 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 $V$.