[nimih]

If you allow reflections of the single tile, an easy counterexample is the pinwheel tiling[1], which is a non-periodic tiling composed entirely of isometric triangles.

An (imo less satisfying) example which does not require reflection of the tile would be as follows: Take a rectangle with width equal to twice the height. You can use this tile to create squares which are either split vertically or horizontally. Put a single horizontally split square at the origin, then tile the remainder of the plane with vertically split squares: this tiling is (rather trivially) not periodic.

[1]https://en.wikipedia.org/wiki/Pinwheel_tiling