A classical result of Hardy and Littlewood states that the uncentred Hardy–Littlewood maximal operator (UHLMO) on balls is bounded on L p (R n ) for all p > 1 and it is of weak type (1, 1). It is well known that, replacing R n with either homogenous trees of degree at least three or the hyperbolic plane, the UHLMO on balls is bounded on L p if and only if p > 2, and it is of restricted weak type (2, 2). Loosely speaking, the different behaviour of the UHLMO on R n and on the hyperbolic plane depends on the fact that (maximal) fam- ilies of balls with large radii have “worse” overlapping properties in the latter case than in the former. In this talk we consider the UHLMO with respect to half balls, and prove that, perhaps surprisingly, this operator is bounded on L p for all p > 1 and it is not of weak type (1, 1) both on homogeneous trees of degree at least three and on the hyperbolic plane. We shall also discuss generalisations of this result to Damek–Ricci spaces. This is joint work with Nikos Chalmoukis (Milano–Bicocca), Effie Papageorgiou (Paderborn) and Federico Santagati (Politecnico di To- rino).