Project Zomboid [Official Site, GOG, Steam] is the impressive Zombie survival game that has single-player and multi-player that's been in development for a long time. This new release adds in quite a lot and vehicles should be next.
Note: The game does not have a SteamOS icon on Steam, but the Linux version is there and it works. The Linux icon is there on GOG.
The latest update, Build 38, adds in a huge Riverside Map which is an equivalent size of the original West Point. It adds in new building types, new tiles and architecture. Here's what the map looks like now:
They've also heavily optimized Zombie placement and movement data for multi-player, which will help to reduce lag and strain on servers.
They've also added in tons more options for when you're playing in the single-player sandbox mode. You can adjust things like how dark it gets at night, if zombies can destroy player-built constructions, fire spread and so on. It sounds like the sandbox mode is a lot more, well — sandbox.
There's a lot more, check out the full changelog here.
The next build should hopefully be the one that finally adds in vehicles to the main stable build. I'm looking forward to that!