Brand new iPad and Apple Pencil can't write on part of screen

Clearly, given your description, the Pencil has been successfully paired with your iPad. Try a forced-restart of your iPad - as this will often clear an isolated software problem:

Force restart iPad (models with Face ID or with Touch ID in the top button)

If your iPad doesn’t have the Home button, do the following:

1. Press and quickly release the volume button nearest to the top button.
2. Press and quickly release the volume button farthest from the top button.
3. Press and hold the top button.
4. When the Apple logo appears, release the top button.

After the forced-restart, you may need to pair your Pencil again with your iPad.

If the problem persists, you would be best advised to visit the Genius Bar at your local Apple Store or Apple Authorised Service Provider (AASP) - where the technicians will be able to hands-on test and assess both your iPad and Pencil. If a hardware fault is identified, you will be in the best possible place to explore your available options:

Genius Bar Reservation and Apple Support Options - Apple

I would use that warranty asap.