  We investigate the statistics of the local time $\mathcal{T} = \int_0^T
\delta(x(t)) dt$ that a run and tumble particle (RTP) $x(t)$ in one dimension
spends at the origin, with or without an external drift. By relating the local
time to the number of times the RTP crosses the origin, we find that the local
time distribution $P(\mathcal{T})$ satisfies the large deviation principle
$P(\mathcal{T}) \sim \, e^{-T \, I(\mathcal{T} / T)} $ in the large observation
time limit $T \to \infty$. Remarkably, we find that in presence of drift the
rate function $I(\rho)$ is nonanalytic: We interpret its singularity as
dynamical phase transitions of first order. We then extend these results by
studying the statistics of the amount of time $\mathcal{R}$ that the RTP spends
inside a finite interval (i.e., the occupation time), with qualitatively
similar results. In particular, this yields the long-time decay rate of the
probability $P(\mathcal{R} = T)$ that the particle does not exit the interval
up to time $T$. We find that the conditional endpoint distribution exhibits an
interesting change of behavior from unimodal to bimodal as a function of the
size of the interval. To study the occupation time statistics, we extend the
Donsker-Varadhan large-deviation formalism to the case of RTPs, for general
dynamical observables and possibly in the presence of an external potential.
