Culture pervades human life and is at the origin of the success of our species. A wide range of other animals have culture too, but often in a limited form that does not complexify through the gradual accumulation of innovations. We developed a new paradigm to study cultural evolution in primates in order to better evaluate our closest relatives' cultural capacities. Previous studies using transmission chain experimental paradigms, in which the behavioural output of one individual becomes the target behaviour for the next individual in the chain, show that cultural transmission can lead to the progressive emergence of systematically structured behaviours in humans. Inspired by this work, we combined a pattern reproduction task on touch screens with an iterated learning procedure to develop transmission chains of baboons (Papio papio). Using this procedure, we show that baboons can exhibit three fundamental aspects of human cultural evolution: a progressive increase in performance, the emergence of systematic structure and the presence of lineage specificity. Our results shed new light on human uniqueness: we share with our closest relatives essential capacities to produce human-like cultural evolution.