Yes. Followers can die permanently from any damage, including friendly fire, unless they are set to “essential”.
When a character set as essential takes nearly fatal damage, they collapse onto their hands and knees (also known as Bleedout) and stumble around for a period of time, instead of dying. [Source — UESP wiki]
Basically, the "essential" status makes followers immune to death from any source.
Also, note that:
All permanent followers become protected NPCs while they are actively following you. When your follower would die due to an enemy attack or a trap, they will be incapacitated for some time and begin to recover their health, just like essential NPCs; enemies will stop attacking or targeting them during this time. However, if they lose all their health again during this period, they will die for good. Thus, you should try to move the fight away from them, so that they are not hit by a stray arrow or area-effect spell. They can die from a damage-over-time effect that is stronger than their health recovery; most commonly those are caused by poisons. [Source — UESP wiki]
The UESP wiki's "Followers" article has a handy list indicating which followers are "essential" and which aren't.
There are ways to prevent “non-essential” followers from dying, such as the Companion's Insight ability. You can get this ability from the Dragonborn DLC by completing the quest “The Winds of Change”. This ability prevents friendly fire from damaging your followers in combat, so you won’t accidentally kill them. Its in-game description is:
Your attacks, shouts, and destruction spells do no damage to your followers when in combat.
Another option for PC players is to use console commands to set a follower as "essential". To accomplish this, follow these steps:
- Enter the following console command:
setessential <base ID> 1. Replace <base ID> with the base ID of your follower. It can be found in your follower’s respective UESP wiki article. In Lydia’s case: setessential A2C8E 1.
- To restore the "non-essential" status, enter the same console command, but enter
0 instead of 1, e.g., setessential A2C8E 0.
There are also mods that can set followers as "essential". PC players can use the UFO - Ultimate Follower Overhaul mod (original/Legendary Edition) / (Special Edition), which lets you set a follower as "essential" through a dialog option.
Xbox One and PS4 players can use the No More Dead Followers mod (Xbox One) / (PS4):
Do you worry about casting a really powerful spell, going berserk with your two-handed weapon, or taking a risky bow shot because your your favorite follower gets in the way? Worry no more! This mod makes all followers (even animal followers) immortal, so they can't be killed--not even by you!
If your follower has died, you may still be able to use console commands (PC only) to resurrect/respawn them. See this Arqade post for more info: How do I resurrect companions / followers?