Assume that $X$ and $Y$ are compactly generated weak Hausdorff spaces (CGWH spaces for short). Assume that they are also well-pointed (so the inclusions of the base points are Hurewicz cofibrations). Is then the mapping space $Y^X$ of pointed maps from $X$ to $Y$ again well-pointed (with base point the constant map from $X$ to the basepoint of $Y$)?

More generally, if $A\to X$ and $B\to Y$ are (unpointed) Hurewicz cofibrations in CGWH, are the natural maps $B^X\to Y^X$ and $B^X\to (Y,B)^{(X,A)}$ cofibrations as well?

This seems to hold if $X$ is compact. Are there more general results?