Ba(Ar) ≈ 750 clusters were generated by associating the supersonic expansion and the pick-up techniques. A femtosecond pump (266.3 nm)-probe (792 or 399.2 nm) experiment was performed to document the dynamics of electronically excited barium within the very multidimensional environment of the argon cluster. Barium was excited in the vicinity of the 6s9p 1P state and probed by ionization. The velocity imaging technique was used to monitor the energy distribution of photoelectrons and photoions as a function of the delay time between the pump and the probe pulses. A complex dynamics was revealed, which can be interpreted as a sequence/superposition of elementary processes, one of which is the ejection of barium out of the cluster. The latter has an efficiency, which starts increasing 5 ps after the pump pulse, the largest ejection probability being at 10 ps. The ejection process lasts at a very long time, up to 60 ps. A competing process is the partial solvation of barium in low lying electronic states. Both processes are preceded by a complex electronic relaxation, which is not fully unraveled here, the present paper being the first one in a series.