What an amazing little game! I'm immediately reminded of old classics like Commander Keen and Dangerous Dave both in style and in level design. I've thoroughly enjoyed my time with this. 

And yeah, you can jump on space.

First of all, the artstyle is great. Gameplay is pretty intuitive and slowly added world features make you wonder what will come next. I really enjoyed playing it. The only thing that I would change are the keybindings (Change jumping from Z to Space).