Solar cycle variations in solar radiation create density changes in any planetary ionosphere, which are well established in the Earth's case. At Mars, however, the ionospheric response to such changes is not well understood. We show the solar cycle impact on the topside ionosphere of Mars, using data from the Mars Advance Radar for Subsurface and Ionospheric Sounding (MARSIS) on board Mars Express. Topside ionospheric variability during the solar cycle is analyzed through neutral scale height behavior. For moderate and high solar activity phases, the topside electron density profile is reproduced with an altitude-variable scale height. However, for the period of extremely low solar activity in 2008 and 2009, the topside was smaller in density than in the other phases of the solar cycle, and there is evidence that it could be reproduced with either a constant scale height or a height-variable scale height with lower electron density. Moreover, the ionosphere during this time did not show any apparent dependence on the EUV flux. This singular behavior during low solar activity may respond to the presence of an induced magnetic field which can penetrate to lower ionospheric altitudes than in other phases of the solar cycle due to the reduced thermal pressure. Numerical simulations of possible scenarios for two different solar cycle phases indicate that this hypothesis is consistent with the observations.