We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate $γ$, the (new) velocity of the particle is drawn randomly from a distribution $W(v)$. We study the survival probability $S(x,t)$ of a particle starting from $x \geq 0$ up to time $t$ and obtain an explicit expression for its double Laplace transform (with respect to both $x$ and $t$) for an arbitrary velocity distribution $W(v)$, not necessarily symmetric. This result is obtained as a consequence of Spitzer's formula, which is well known in the theory of random walks and can be viewed as a generalization of the Sparre Andersen theorem. We then apply this general result to the specific case of a two-state particle with velocity $\pm v_0$, the so-called persistent random walk (PRW), and in the presence of a constant drift $μ$ and obtain an explicit expression for $S(x,t)$, for which we present more detailed results. Depending on the drift $μ$, we find a rich variety of behaviours for $S(x,t)$, leading to three distinct cases: (i) subcritical drift $-v_0\!<\!μ\!<\! v_0$, (ii) supercritical drift $μ< -v_0$ and (iii) critical drift $μ=-v_0$. In these three cases, we obtain exact analytical expressions for the survival probability $S(x,t)$ and establish connections with existing formulae in the mathematics literature. Finally, we discuss some applications of these results to record statistics and to the statistics of last-passage times.