If you want to make this believable I would draw heavily from existing large Player Vs Environment games, just tweak the fluff/justification around the existing gameplay.
These game exist, and are huge. Most obvious is World Of Warcraft. Yes it has PVP elements, but one can play on a PVE server without every fighting a human enemy. Instead you have a group of 6, or 10, or 25, players all working together to take on a challenge the game provides to them. Harder challenge comes from taking on a stronger dungeon/raid etc.
Or look at MOBA's (leauge of Legends, Heroes of the Storm, Defense of the ancients etc). Sure, the standard MOBA is pretty much the definition of competitive. However, it's possible for your 5 man team to go up against AI foes in a player vs environment setting. Sure, standard MOBAs the Ai are not that good and are kind of easy, because their just training tools for the real game, but a game that takes the same format of a MOBA, 5 man teams in a 20-45 minute game against a computer controlled challenge, can easily be created.
Now I know you're first thought is obvious, these are clearly non-peaceful games, and that's true. However, the gameplay is not at all competitive, and does not need competition. It is quite easy to keep the gameplay and just modify the justification for what the characters are doing. Your find much of design elements carry over quite easily, and should be used as a model for your game. I would suggest tweaking characters into being heroes fighting some disaster or external/environmental threat most often.
WoW for wildlife management
Your exploring new lands (either new planets in a sci-fi game or a fantasy game with some ability to jump to new unknown locations though some magic gate recently discovered). These lands have large and potentially hostile creatures. You need to explore these lands to learn their secrets, discover any resources your people can use, etc.
Along the way you need to deal with hostile creatures & dangerous land elements. You don't attack the creatures, but have a whole host of abilities to calm fearful creatures, charm others that you need to get close to study, protect against environmental hazards that show up, etc.
Your dungeon is effectively a new world to explore and your doing it meaning no harm. In fact take it a step further, perhaps the creatures are being harmed by something (magic invading the world is driving them insane, or space plague risks killing them) so you need to actively hunt down and control the creatures precisely so you can help provide them with whatever aid will protect them from the disaster that recently struck their land.
MOBA for flying a spaceship
Imagine you have a game set in the future where you have a spaceship doing some important space-mission (could be anything from studdying anomolies in dangerous space to trying to survive a space jump into new unexplored area whatever). Your 5 man team is in charge of various roles on the spaceship, and need to cooperate to make the ship work it's best to survive the challenge. Each character has a set abilities they can use but need to collaborate with each other to get the most out of them. Alternatively to make this feel more like a MOBA modify it to be a space fleet and each character controlls one specialized craft that needs to work together to protect the rest of the civilian/scientific/escort-quest ships in the dangerous mission their all working towards.
Team Fortress for fighting wildfires
Imagine a crazy wildfire spreading and people needing to combat it. You create a team of 'elite specialized firefighters' that get airdroped into a disaster zone who help to combat the blaze. Your "heavy" is the guy with a giant firehose that can put down invading flames rapidly, Your 'spy' is someone who's job is to run into the burning building to grab civilians that need saved, with only a small anti fire extinguisher but special abilities to survive being surrounded by fire. Your Engineer builds fire hydrants that others can connect to for increased water pressure and automated water turrets. Your Medic checks out everyone gear and helps cool them off when their running dangerously high in their 'overheat' ranges.
The game would have to make fire a little more dynamic, but there are all kinds of justifications I could come up with to do it, or none at all (a FPS isn't all that realistic compared to a real battleground, people don't mind your taking liberty to make the game fun)
4X games (Xplore, Xpand, Xploit, Xterminate) games can easily be built into 3X games (drop the extermination part) with little effort at all. Everyone can be in charge of building up their own civilization without combat involved. Perhaps they all cooperate together and are ranked not just on how their own civilization grows and develops but also how others grow as well.
All you really need to do here is add more environmental hazards to the game, more random events, famine, diseases, negative weather patterns etc, to add extra pressures to fledgling civilizations to make it hard for them to grow, perhaps combined with some extra elements to ensure that banding together usually helps the civilizations to survive these hazards.
I think I sort of made my point so I won't keep going on. These are examples in general, there are plenty of other ways of doing the sort of things I suggested. My main point is rather then trying to reinvent the wheel I suggest taking existing games and just throw on a non-violent skin. The challenges really don't need violence and the gameplay can work pretty similarly without violence.