Consider the 3-d primitive equations in a layer domain Ω = G × (−h, 0), G = (0, 1) 2 , subject to mixed Dirichlet and Neumann boundary conditions at z = −h and z = 0, respectively, and the periodic lateral boundary condition. It is shown that this equation is globally, strongly well-posed for arbitrary large data of the form a = a 1 + a 2 , where a 1 ∈ C(G; L p (−h, 0)), a 2 ∈ L ∞ (G; L p (−h, 0)) for p > 3, and where a 1 is periodic in the horizontal variables and a 2 is sufficiently small. In particular, no differentiability condition on the data is assumed. The approach relies on L ∞ H L p z (Ω)-estimates for terms of the form t 1/2 ∂z e tA σ Pf L ∞for t > 0, where e tA σ denotes the hydrostatic Stokes semigroup. The difficulty in proving estimates of this form is that the hydrostatic Helmholtz projection P fails to be bounded with respect to the L ∞ -norm. The global strong well-posedness result is then obtained by an iteration scheme, splitting the data into a smooth and a rough part and by combining a reference solution for smooth data with an evolution equation for the rough part.2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 47D06, 76D03, 86A05.