Finishing the mission without killing anyone will unlock the “Pacifist” trophy/ achievement. To do this, you’ll have to use the Uragan pistol which fires non-lethal rounds that will instantly knock out any soldier it hits. It doesn’t take out vehicles so you have to aim for the driver in order to stop them. To know how to do this, refer to the videos below:
You have to take note of the following:
Once the mission starts. switch to your non-lethal pistol and start shooting at targets. Your ammo will be unlimited so just shoot away until you spot the target's flare. Once your chopper flies over the prison camp, prioritize taking out the soldiers with rocket launchers. As your VIP drives away, take out the pursuing jeep and the enemies shooting at you. As he makes his turn, a military truck will block his way. Aim for the driver to prevent the truck from pursuing him.
Continue protecting the VIP from shooters and pursuers. On your way to the helipad, the enemies will man the AA gun. Take out the gunner and continue shooting enemies as much as you can. It will be challenge to hit enemies at a long distance so do your best in adjusting your shots accordingly. Once your target reaches the helipad, he will lose control and his jeep will crash. As your chopper maneuvers itself to reach the target, more enemies will appear. Be careful of the rocket launcher-equipped infantry on the Central Tower. Continue clearing the area of enemies until you get the cue to retrieve the VIP on foot.
Before picking up the VIP, get the rocket launcher nearby. Carry the VIP and an armored vehicle will appear. Quickly place the VIP behind an indestructible cover then lure the armored vehicle away from the knocked out soldiers, to prevent them from getting caught in the rocket's splash damage, which will ruin your chance in unlocking this achievement/ trophy. If you want to play it safe, you can run to the AA emplacement nearby and use it to shred the armored vehicle to pieces.
Once done, take out the enemy reinforcements and board the chopper with the VIP. An enemy chopper will pursue you. You can use the rocket launcher to take it out; however, make sure that the rocket won't hit the gunners on the window. If you want, you can still use the Uragan pistol to take out the gunners. You won't destroy the chopper but it will stop pursuit and the mission will be complete.