[contravariant]

Brouwer's fixed point theorem only applies to compact convex sets.

Infinite lines don't work, as they are not compact.

Similarly a circle would not work as it is not convex (you're close with your example, you just need to glue together the endpoints to turn it into a circle and make the map continuous).

[d_tr]

The line example is what I thought and then I looked the theorem up in Wikipedia. Thanks for pointing this out.