Yes, it is consistent. The following theorem is proved by Shelah and myself (our  paper  is not yet complete):

>**Theorem.** Assume $\kappa$ is weakly compact and $\lambda > \kappa$ is 2-Mahlo. Then there is a generic extension in which $\kappa=\aleph_2, 2^{\aleph_1}=\aleph_3=\lambda$ and any forcing notion which adds new subset to $\aleph_2$,  collapses $\aleph_2$ or $\aleph_3.$

Unfortunately the proof is not so easy to state it here.

Note that by a new subset to $\aleph_2$ I mean a subset of $\aleph_2$ which is not in $V$, the ground model, but such that all its initial segments are in $V$.

>**Remark.** It is evident that the above theorem gives the consistency of the requested question. If the forcing does not add subsets to $\aleph_1,$ then it adds new subsets to $\aleph_2$ so the theorem applies.

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**Added note:** The paper is now available at [Specializing trees and answer to a question of Williams](https://www.researchgate.net/publication/319015756_Specializing_trees_and_answer_to_a_question_of_Williams).