The relaxation times of excited states of arsenic dopant in germanium at cryogenic temperatures T < 15 K have been experimentally studied by the optical pump-probe method using radiation of a free-electron laser. Two variants of the excitation of impurity centers have been used in the experiment: (i) from the 1s(A1) ground state of the dopant and (ii) from the 1s(T2) first excited state having a finite thermal population. In the former variant, it has been shown that the decay times of the 2p0 and 3p± states are about 0.8 and 0.6 ns, respectively. In the latter variant, a single measurement can simultaneously provide the relaxation times of two 2p± and 1s(T2) states about 0.6 and no more than 0.16 ns, respectively. The data obtained have indicated the possibility of forming population inversion and the gain of terahertz radiation at the 2p± → 1s(T2) and 2p0 → 1s(T2) transitions at the optical excitation of the mentioned impurity centers.