People have pretty much hit on the basic problem with your question already and provided some workarounds. Your problem is you usually need competition to be a game. But for each challenge you can tweak it so that chances for inter-personal conflict are reduced. Your challenge types are:
Avoiding PvP is one of your goals and as others have stated an algorithm can take the place of the enemy player. An adaptive algorithm or AI that takes into consideration play from the group's previous rounds would be one that got progressively more difficult. One example of an "algorithm" would be the game rules or for an "AI" the GM/DM in an RPG. Establishing either of these turns a PvP into a PvE. PvE is rather simple especially if there's a random factor. You almost never have negative competition between players unless there's a scoring metric that applies to all of them.
For example if your playing a game where you're each a collector and you get a point for collecting an item of your own type but none for collecting another players'. If you had each player score individually it would be competitive but if you had the team score be the sum or the fact that every collector found something, then you have a game where everyone wants to help other players find their collectable because it increases team score/win. The only score is if the team wins or not and/or by how much. You could have competition between larger groups but that's a different game at that point (you have a different scoring system). If you kept the collaborative scoring of the smaller groups it just becomes a larger game.
Another example is a "Validating" game. There's a second win condition that can only be met by losing so both players win, but in different ways. Take a war simulation in an army. The winner wins and the loser proves a point that helps the team (they're all on team army). The point if the enemy is the loser is we're doing a good job we think. The point if the team is the loser is that we have things we need to fix together.
Ways to make PvP non-competitive in a sense: make scores binary (if you add a factor to ensure winning this results in scores of 1:1), make team score a composite of personal scores (Collaborative), make scores for each player orthogonal (PvS), make competition game rules or AI (PvE), have a orthogonal secondary goal that is met only by losing ("Validating"), have a never-ending PvP game (No Score Game).
PvE has only the problem of losing. We can transform any such problem rather easily though. If the team would lose start a "Fate Game". This saves the player from their immediate doom at the cost of time. For example: start a new game that is slightly easier. Fate Games can have Fate Games inside themselves. Eventually your chance of losing reaches 0 and you clear the loss in the main game. The better you are as a team the quicker the game. There are of course other ways to transform a game with losses into one without losses.
PvS is also in the category of creative games. PvS are "non-competitive" because they're self-competitive by definition. Your goal or your team's goal is to do something they couldn't have done before. Either yourself overcoming your expectations or the group outperforming an individual. Think tanks and Exquisite corpse fall into this category. Once again, as long as your playing the base PvS game nobody loses. Creative games can also be outside of this category; Mao, Nomic, Mornington Crescent, and 1000 Blank White Cards are in general PvP creative games. Those types of games by their very nature of modification are only valid if there are "super-rules" which say everyone must win. The challenge then becomes ensuring this.
Creative games are a wild card, that in general can be solved by adding your conditions as super-rules.
To meet your goals the game only needs to get harder which is something that, if culturally understood, is inevitable. If you're guaranteed a win given enough time then the goal is to have fun. A narrow (and yet broad) definition of fun is a challenge that is surmountable. For a really narrow look at this definition: The more the challenge, the more the sense of accomplishment. The more surmountable, the quicker you get the reward. Goal is to minimize time and maximize challenge. So if the reason everyone is playing is to have fun it makes sense that they'd make the game harder in a PvS. You could even have an assigned antagonist. You would think their goal would be to win, but if the super-rule is everyone must win they have the challenging task of making the game winnable but as hard as possible. So their unspoken win condition is "lose, but make it a show". In a PvE you can evolve the game based on player performance. And in a PvP you just make it into a PvE or PvS (or make it never end).