Played for a while and was enjoying myself. The battle system reminds me of the old Zelda game where you can only attack (or in this case point your sword) in the moving direction. It's a quirky but quite nostalgic battle system that I really love. The idea of invincible enemies is interesting too, but the game does spawn too little of them to make an impact on the gameplay. I would suggest a lot more of them, but maybe make them stand out a bit to reduce the frustration of the players.
I am not sure if this is a bug or a feature, but I felt like the player can't be damaged really ruined the tension of the game, and made the health pickups quite useless. I think that with that fix, the game will be more exciting. Sadly, the game ends in the snowy area quite abruptly, and as other commenters said, it would be nice to have a boss fight as the final climax of the game.
Anyway, good job in making this game. Keep up the good work!