This is a follow-up of this question. Recall that a metric space $(X,d)$ is called a path-metric space if the distance between any two points in $X$ equals the infimum of lengths of paths between these points. Such $(X,d)$ contains a plenty of rectifiable paths (unless $X$ is either empty or is a one-point set). A geodesic between the two points $x, y\in X$ is a path $c$ from $x$ to $y$, whose length equals the distance $d(x,y)$.
Question. Is there a complete path-metric space $(X,d)$ (consisting of more than one point) such the only geodesics in $(X,d)$ are constant maps to $X$? (I think so, but do not see a clear proof.)
Obviously, such space $X$ cannot be locally compact (at any point).
Note that in the example given in the linked question, there are pairs of points without geodesics connecting them. Another example (an infinite-dimensional complete Hilbert manifold) is mentioned in this wikipedia article. However, in both examples, the space contains plenty of geodesics.
Note. For the record: I asked this question first here.
Edit. I just discovered that the same question was posted by Anton Petrunin (in 2010) and answered (by fedja) on Mathoverflow here. Sadly, the only way to find out was to post this question and then to check the list of similar questions. Feel free to close as a duplicate.
