We have investigated the electron and hole spin dynamics in p-doped semiconductor InAs/GaAs quantum dots by time resolved photoluminescence. We observe a decay of the average electron spin polarisation down to 1/3 of its initial value with a characteristic time of T<sub>Δ</sub> ≈ 500ps. We attribute this decay to the hyperfine interaction of the electron spin with randomly orientated nuclear spins. Magnetic field dependent studies reveal that this efficient spin relaxation mechanism can be suppressed by a field in the order of 100mT. In pump-probe like experiments we demonstrate that the resident hole spin, "written" with a first pulse, remains stable long enough to be "read" 15ns later with a second pulse.