Has the iron core of Mars really solidified? In the Nova episode "Origins: Earth is Born", Neil DeGrasse Tyson states, "But Mars is just a fraction the size of the Earth, so it cooled more rapidly. And as it cooled, its molten iron core hardened. As a result, Mars stopped generating its magnetic shield." (February 28, 2004)

see: http://www.pbs.org/wgbh/nova/space/origins-series-overview.html#origins-earth-born

This comment from Tyson is consistent with what I have long believed. But I wonder.

The melting point of iron (Fe) is around 1538 °C.

see: https://en.wikipedia.org/wiki/Melting_points_of_the_elements_(data_page)

The the geothermal gradient for Mars is estimated to be around 6 °C per km.

see: What is the temperature 55 km beneath the surface of Mars?

If the estimate of 6 °C per km is the ballpark, then I would expect that iron is molten as a mere 256 km of depth. At the core of Mars (over 3000 km in depth) I would expect the temperatures to be far higher than 1538 °C.

It seems to me that it is impossible for iron to be in solid form at Mars' core.
You're mostly missing an important point to do with the geothermal gradient.

The geothermal gradient used in that question, and applied to a $55\:\mathrm{km}$ depth is not consistent all the way through Mar's interior. They cite the Earth's geothermal gradient (see image below) and from that page you can see how the temperature profile changes with depth. There is a much stronger gradient towards the surface. Thus you can't naively apply the surface geothermal gradient through to the core. Rather, to get the core temperature, you'd need to integrate the gradient function.

Earth's Temperature Profile