This creature exists in an infinite world, a flat landscape that extends ad infinitum, where light rains from the sky from infinity during recurrent day-night cycles, and, similarly, the ground goes down continuously.
All kinds of critters populate the cosmos including the skies and the underground realm. This bizarre world has existed for an infinite time and for this reason I want (at least) one creature to have inhabited it for an infinite time. And that's not all, this particular being is also physically infinite(its body has no end); obviously I don't want it to take up all the space, but only a part of it.
In the specific I was thinking of giving it a shape similar to something like Gabriel's Horn, so it would leave a large area available, however this would basically cut most of the surface in half, and on top of that most of his body is vulnerable to attack since it's so thin; for this reason I would like to avoid "infinitely small" bodies.
What infinite shape can I give my organism so that it wouldn't be too vulnerable but at the same time leave room for people to move about relatively easily, without cutting off parts of the world that are close by from each other? Also, while its body is infinite it should have only one head.
I am going to be a bit lax with the physics of this world to allow for some basic functionalities, but I want to put special attention to this aspect.
EDIT: Thanks a lot to everyone for the inputs, I went with the answer that better fits the requirements, but I will probably incorporate elements from other interesting answers