Experiments were carried out in 2018 and 2019 to evaluate the effects of a biostimulant of animal origin, with a hormone-like action, on the growth and physiology of young olive trees. The biostimulant, consisting of a complex of natural amino acids (glycine, proline, and hydroxyproline, etc.), was applied by fertigation to both potted (one-year-old) and field-grown (2 years after transplanting) young olive trees. The control consisted of trees treated with an amount of urea nitrogen equal to the total nitrogen supplied with the biostimulant. Potted trees treated with the biostimulant showed greater growth than the control soon after treatment and up to the end of the vegetative season. Generally, this was associated with higher leaf photosynthetic rates and stomatal conductance. At the end of the experiment, all tree parts (roots, stem, shoots, and leaves) of the treated trees had greater biomass than the control. The trunks of field-grown trees treated with the biostimulant had greater growth with respect to the control. Overall, the results indicate the possibility of using the biostimulant based on complexes of natural amino acids to promote the growth of young olive trees.