Is there a smooth homotopy lifting theorem?

Question

Given a smooth map $p\colon E \to B$ of smooth manifolds with the continuous homotopy lifting property, does $p$ satisfy the smooth homotopy lifting property?

Answer 1

A trivial counter-example to this is the map $p(x)=x^3$, $p: E={\mathbb R}\to B={\mathbb R}$. It is a homeomorphism, hence, satisfies the continuous homotopy lifting property (HLP). To see that it fails to have the smooth HLP, consider the maps $f_0(x)=0, f_1(x)=x$ and the (smooth) linear homotopy $f_t$ between the two. The map $f_0$ of course lifts to the constant map $\tilde{f}_0(x)=0$ with respect to $p$. On the other hand, the map $f_1$ does not have a smooth lift (the only possible lift would be $\tilde{f}_1(x)=x^{1/3}$ which is not smooth). Hence, the smooth HLP fails for the map $p$.