Let $X$ be a contractible space. If $x_0 \in X$, it is not necessarily true that the pointed space $(X,x_0)$ is contractible (i.e., it is possible that any contracting homotopy will move $x_0$). An example is given in 1.4 of Spanier: the comb space. However, this space is contractible as a pointed space if the basepoint is in the bottom line.
Is there a contractible space which is not contractible as a pointed space for any choice of basepoint?
My guess is that this will have to be some kind of pathological space, because for CW complexes, we have the Whitehead theorem. (So I'm not completely sure that the Whitehead theorem is actually a statement about the pointed homotopy category, but hopefully I'm right.)