Let X be a nowhere vanishing complete vector field on a manifold M, $\gamma: \mathbb{R} \to M$ be its flow with $\gamma(0)=p \in M$ and suppose it is not periodic. If $\gamma(\mathbb{R})$ is closed, is $\gamma$ a proper map?

The only counter-example I can think of is a curve looping back to itself such that $\lim_{t\to + \infty} \gamma(t) = \gamma(\hat{t})$ for some $\hat{t} \in \mathbb{R}$ but I don't know if this can arise from a nowhere vanishing complete vector field.