Whilst not discounting the possibility of a dead Pencil battery, there appears to be a bug affecting some users of iPad Air3 that interferes with pairing of the Apple Pencil. Another contributor reports having found a workaround solution.
“... can confirm the issue is iOS 13.5. The only way to pair your 1st Gen Apple Pencil to your iPad Air 3 is to quickly connect and then disconnect power to your iPad then plug the Pencil into the Lighting port. This will fix the issue and will pair successfully. It’s like the quick jab of power to the iPad ‘wakes’ some part of the software into action. I can confirm I’ve tried this on 2 different iPad Airs with same result.”
This discussion details the workaround:
https://discussions.apple.com/thread/251382715
If you have an iPad Air3 and have recently updated to iPadOS 13.5.1, this may be your issue.
Otherwise, as described, your Apple Pencil is displaying classic symptoms of a failing or dead battery - that will not accept or hold any charge.
Assuming that you do not have any other iPad charging issues - and that the Lightning Port of your iPad is free from debris - whilst the Pencil is plugged-in to the Lightning Port of your iPad, verify that the Pencil is recognised:
Settings > General > About
Scroll to the bottom of the page - and you should find two identical entries for the Pencil (one each for the physical and Bluetooth connections).
The tiny internal Li-ion battery is susceptible to permanent/irreversible damage through being left discharged for long periods. Even some “new” pencils can exhibit signs of failure out-of-the-box if they are “old stock”.
It is essential that if you have an Apple Pencil that you charge it regularly - whether used or not - so as to protect the battery from deep-discharge. Do not allow a pencil to remain in low-charge state for any period of time - as the internal battery will fail, rendering the Pencil useless.
If the Pencil Battery has failed, the only remedy is to replace the Pencil. If purchased directly from Apple, if the battery has failed and your pencil is within its one-year warranty, you should look to having it replaced when the Apple Stores eventually reopen.
If the above possibilities are eliminated, here is one of the better troubleshooting guides for Apple Pencil:
https://appletoolbox.com/apple-pencil-not-working-heres-our-troubleshooting-guide/
I hope this information proves to be helpful in both diagnosing and resolving the problem.
