No. The following theorem is from a work in progress by Yair Hayut and myself.

Theorem. If $\Bbb P$ is a proper forcing, and it changes the cofinality of $\kappa$ is $\mu>\omega$, then $\Bbb P$ adds a surjection from $\mu$ onto $\kappa$.

Now suppose that you had such countably closed $\Bbb P$, it is certainly proper. And it changes the cofinality of $(\lambda^+)^V$ to be something which is smaller than $\lambda$, and therefore collapses $(\lambda^+)^V$, and as a consequence it must collapse $\lambda$ as well.

(It should be remarked that a countably closed forcing cannot change the cofinality of something $\omega$ anyway.)

No. The following theorem is from a work in progress by Yair Hayut and myself.

Theorem. If $\Bbb P$ is a proper forcing, and it changes the cofinality of $\kappa$ to $\mu>\omega$, then $\Bbb P$ adds a surjection from $\mu$ onto $\kappa$.

Now suppose that you had such countably closed $\Bbb P$, it is certainly proper. And it changes the cofinality of $(\lambda^+)^V$ to be something which is smaller than $\lambda$, and therefore collapses $(\lambda^+)^V$, and as a consequence it must collapse $\lambda$ as well.

(It should be remarked that a countably closed forcing cannot change the cofinality of something $\omega$ anyway.)

No. The following theorem is from a work in progress by Yair Hayut and myself.

Theorem. If $\Bbb P$ is a proper forcing, and it changes the cofinality of $\kappa$ is $\mu$, then $\Bbb P$ adds a surjection from $\mu$ onto $\kappa$.

Now suppose that you had such countably closed $\Bbb P$, it is certainly proper. And it changes the cofinality of $(\lambda^+)^V$ to be something which is smaller than $\lambda$, and therefore collapses $(\lambda^+)^V$, and as a consequence it must collapse $\lambda$ as well.

No. The following theorem is from a work in progress by Yair Hayut and myself.

Theorem. If $\Bbb P$ is a proper forcing, and it changes the cofinality of $\kappa$ is $\mu>\omega$, then $\Bbb P$ adds a surjection from $\mu$ onto $\kappa$.

Now suppose that you had such countably closed $\Bbb P$, it is certainly proper. And it changes the cofinality of $(\lambda^+)^V$ to be something which is smaller than $\lambda$, and therefore collapses $(\lambda^+)^V$, and as a consequence it must collapse $\lambda$ as well.

(It should be remarked that a countably closed forcing cannot change the cofinality of something $\omega$ anyway.)