As of the 2019-06-18 "Back & Kickin’ Brass" update, perks can be prestiged up to rank 5.
When you Prestige a Perk, you reset it back to Level 0, losing access to all of its Perk Bonuses and Perk Skills until you level it back up. In exchange you are given a special Weapon Skin, along with a modified Perk Icon and 2,000,000 Vosh (enough for 4 Vault Crates). You are also given an Experience boost so leveling up the next time will go by faster.
There are (currently) no Achievements tied to Prestiging, and the rewards are purely cosmetic so the feature can be ignored if you so choose.
Rank 1 Prestige Rewards
- Experience Bonus: 30%
- Weapon Skin: Knives for each Perk
Rank 2 Prestige Rewards
- Experience Bonus: 35%
- Weapon Skin: Tier 1 Weapons for each Perk
Rank 3 Prestige Rewards
- Experience Bonus: 40%
- Weapon Skin: Tier 2 Weapons for each Perk
Rank 4 Prestige Rewards
- Experience Bonus: 45%
- Weapon Skin: Tier 3 Weapons for each Perk
Attempting to prestige past the maximum level will present you with the error message, "Max perk prestige level achieved!"