Assume there is a radioactive material which emits particles according to a Poisson process at rate $\lambda$. Each particle stays alive for 1/μ time units (deterministic time).
Define $X=\{X_t,t\geq0\}$ to be the number of live particles at time $t$.
Is $X$ is Markov jumping process? Does the fact that the lifetime of particles is deterministic (and not exponentially distributed) cause a problem?
You're right that the deterministic transition creates a problem and $X$ is not a Markov jump process.
It does not satisfy the Markov property because the future is not dependent only on the current state, that there are $n>0$ live particles, it also depends on the past (specifically the emission times of each of the $n$ particles).