The first Geneva Convention, adopted by Prussia amongst others in 1864, makes specific provision for the protection of sick and wounded personnel including those treating them. Updated in 1906 (to cover maritime matters) and 1929 it would have been well known to soldiers on both sides that it was illegal to target medical facilities, medical personnel or soldiers considered hors de combat.

Although this is rather more than a Gentlemen's agreement, the principle that enforces adherence is similar - if you go around shooting the enemy's medics then you are inviting the enemy to shoot your medics.

In it's present form, the convention protects all medics, medical facilities and patients as well as making it an act of perfidy to use medics or medical facilities for combat operations. Any medic or medical facility used for combat operations becomes a legitimate target and any injured soldier engaging in combat loses their right to protection.

Tell me about this football match

Edit: sorry that I wasn't aware about this match that it forsaken me with down votes

There was no official or even a mutually agreed under-the-table agreement not to shoot Medics. Medics helped healthy men to fight and medics could fight back if needed (regardless of their official position, which has altered over time and from army to army). Although the Geneva convention forbade attacking medical facilities or their crew this wouldnt necessarily have bled down to front line medics. Any medic running out under fire MAY get an easier time than a non-medic, but it would require ALL the opposing troops to recognise his status or intentions and then decide to honour the spirit of that. This naturally could vary enormously.

So just because it worked once, doesnt mean you're notgoing to get riddled with bullets the next time.