Properties of loops under lifting

Question

Let $p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0) $ be a covering space. Is it always true that if the image of a path $\tilde{\gamma}$ under $p_*$ is a loop $\gamma$ based at $x_0$, then $\tilde{\gamma}$ is a loop based at $\tilde{x}_0$? Do we have to make connectivity assumptions about the spaces $X$ and $\tilde{X}$?