You can in rare cases get nerve damage from sleeping on your arm.
However getting nerve damage from sleeping on your arm once in awhile is unlikely to cause any actual nerve damage whether or permanent or temporary.
Sleeping on your arm can cause temporary nerve compression and cause tingling and numbness which is also known as the arm falling asleep and is a normal occurrence and happens to many people.
Most people are just fine after sleeping on their arm the arm numbness often goes away within a few minutes or so of waking up and moving your arm around.
However prolonged or repeated pressure on the nerves and especially your radial nerve can cause more serious issues such as wrist drop.
Your body will also wake you up before any actual damage to your nerves sets in so you should be just fine.
I have an arm falling asleep at least once a week if I roll over on it or have the arm placed under a pillow and often times I'm so asleep that I don't even know I'm in these positions.
When we sleep, we often unintentionally place pressure on the nerves and most commonly the radial nerve.
The radial nerve is a nerve that runs from the armpit to the hand.
The pressure can restrict the blood flow and cause temporary numbness, tingling or a pins and needles sensation which goes away pretty quickly in most cases.
Occasionally placing pressure on your arms and your arm going numb is unlikely to cause any issues but in rare cases it could lead to nerve damage such as Saturday night palsy or honeymoon palsy which can occur when you fall asleep when a nerve is compressed for a long period which can lead to wrist drop.