Let $M$ be a compact Riemannian manifold, $\bar M \to M$ be its universal covering and $\phi \in Isom(M)$ be an isometry of $M$. Is it true that, if $\phi$ is isotopic to the identity map of $M$, than (any or some) its lift $\bar \phi \in Isom(\bar M)$ has bounded distance to the identity map of $\bar M$?
A motivation is that the statement for dimension 2 is mentioned in Farb, Margalit "A primer on mapping class groups", version 5, page 215, and I can not prove it either.
Yes, it is.
Let $\Phi$ be an isotopy. That is $\Phi:I\times M\to M$ is smooth, such that $\Phi(0,p)=p$ and $\Phi(1,p)=\phi(p)$. Let $\overline{\Phi}:I\times \overline{M}\to\overline{M}$ denote the lift of $\Phi$ equal to the identity at time $0$. By compactness, there is some $\lambda>0$ such that for every $(t,p)\in I\times M$ we have $$\left|\frac{\partial\Phi}{\partial t}(t,p)\right|<\lambda.$$ Since the covering map is a local isometry, it follows that we also have $$\left|\frac{\partial\overline{\Phi}}{\partial t}(t,\overline{p})\right|<\lambda$$for every $(t,\overline{p})\in I\times \overline{M}$, and the claim follows.