Let $E$ be a normed $\mathbb R$-vector space, $(X_t)_{t\ge0}$ be a nontrivial $E$-valued càdlàg Lévy process and assume $c>0$ with $\|\Delta X_t\|_E\le c$ for all $t\ge0$. Let $$\tau:=\inf\{t\ge0:\|X_t\|_E\ge c\}.$$
Are we able to deduce that $\tau<\infty$; maybe at least almost surely?
Clearly, if $t\ge0$, then $\{\tau\ge t\}=\{\forall s\in[0,t):\|X_t\|_E<c\}$.