If a character has fallen into evil, and does not wish to repent, the paladin can do anything to them.
Paladins are God's Hand upon the earth. If the paladin leaves an evil character alone, then the paladin condemns innocent victims to be hurt by that character. If a paladin encounters evil, the paladin must deal with it.
How can a paladin deal with it? Well, the preferred option is to redeem them back to the path of Good. Failing that, imprison them. Failing that, kill them.
A paladin does not let them walk free.
God has granted paladins the power to detect evil, and Evil is real. It's not "guy having bad thoughts", it is the cumulative result of many, many evil deeds.
The only time a paladin can leave an evil creature free is if a greater evil must be stopped, and on many such occasions I suspect that atonement must be sought (especially if the paladin had to work with the evil character).
The paladin will prefer to redeem the evil character, but if that is impossible because of time restraints, the paladin will imprison that character.
If the paladin cannot imprison the evil character, then there is no option left but to slay the character. It is regrettable, but necessary to uphold Good upon the land.
Cheers!