As far as I know, it is an open problem whether there exists a single connected tile that tiles the plane only non-periodically.
Is the situation different for a strip? (Or, for that matter, a half strip, or quadrant, or bent strip.) I mean, is it possible for a single tile to tile a strip only non-periodically (or half strip, quadrant, bent strip), or is it also unknown?
(I ask generally, but I am particular interested in tiles that are polyominoes.)
To give some context: I'm on a self-study adventure of polyominoes and their tilings especially (I asked some previous questions that up came up in this context: Does every domino tiling have at least two “exposed” dominoes? and What rectangles can a set of rectangles tile?). This latest one comes from trying to see which strips (and half strips etc.) a polyomino can tile. One way to do this would be to show that you cannot form any parallelogram-like* pieces that can can be joined repeatedly to form the strip (or half strips etc.) But this will only work if strips (etc.) can always be tiled periodically if at all.
(*) Parallelogram-like piece = a piece that has four "sides"; two opposites that are straight and the same length, and the other two no necessarily straight but the onbe is a translate of the other. For polyominoes, it's figures where all rows have equal number of cells and they are all row-connected. (I also asked here if there is a term for such polyominoes: Is there a word for a polyomino with $n$ connected cells in each row?. I did not get an answer, and after looking through hundreds of papers I haven't found one.) Here are some examples.
(Clearly such can be put together to tile a strip.)