Let $X$ be a 1-bounded ultrametric space, i.e. a pair $(|X|,d_X)$ where $|X|$ is a set and $d_X:|X|\times |X|\to [0,1]$ is a distance such that $d_X(x, x'')\le \max\{d_X(x,x'),d_X(x',x'')\}$. The category of 1-bounded metric spaces and non-expansive maps is cartesian closed, where a non-expansive map $f:X\to Y$ is a function such that $d_Y(f(x),f(x'))\le d_X(x,x')$. Every non-expansive map is continuous, but the converse need not hold.
Is the category of 1-bounded metric space and continuous functions cartesian closed? I didn't find an answer to this, although it seems to me that the answer should be negative. But my only argument is that a positive answer would give a cartesian closed full subcategory of TOP that i should have been able to find somewhere.