We revisit the classical problem of market impact through the lens of a new agent-based model. Drawing from the mean-field approach in Statistical Mechanics and Physics, we assume a large number of 'agents' interacting in the order book. By taking the 'continuum' limit we obtain a set of nonlinear differential equations, the core of our dynamical theory of price formation. And we explicitly solve them using Fourier analysis. One could talk as well of a "micro-macro" approach of equilibrium, where the market price is the consequence of each ("microscopic") agent behaving with respect to his preferences and to global ("macroscopic") information. When a large market order (or "metaorder") perturbs the market, our model recovers the square-root law of impact, providing new insights on the price formation process. In addition, we give various limiting cases, examples and possible extensions.