Transition-metal dopants such as Mn determine the ferromagnetism in dilute magnetic semiconductors such as Ga1-xMnxAs. Recently, the acceptor states of Mn dopants in GaAs were found to be highly anisotropic owing to the symmetry of the host crystal. Here, we show how the shape of such a state can be modified by local strain. The Mn acceptors near InAs quantum dots are mapped at room temperature by scanning tunnelling microscopy. Dramatic distortions and a reduction in the symmetry of the wavefunction of the hole bound to the Mn acceptor are observed originating from strain induced by quantum dots. Calculations of the acceptor-state wavefunction in the presence of strain, within a tight-binding model and within an effective-mass model, agree with the experimentally observed shape. The magnetic easy axes of strained lightly doped Ga1-xMnxAs can be explained on the basis of the observed local density of states for the single Mn spin.