FirstPersonController can walk through Mountains

I have set up a very simple project:

• I added a FPSController from the Prototyping-Assets

• I created a Terrain Gameobject

• I created a mountain with the "Raise/Lower Terrain"-tool

I didn't modify anything else!

When I start the game, I can walk on the terrain, but the FPSController does not collide with the mountain. It simply runs through.

I found some threads in several forums on similar problems, but the solution always was: Make sure your Gameobject has a Rigidbody-Component and a Collider; the default FPSController has that! So what else could be the reason for that weird behaviour?