As the OP pointed out, my original answer was wrong. In fact centralizers of elements _are_ unimodular, and a reference can be found by following the link in the OP's comment below.
Here is an example that I find interesting: Consider the element $\begin{pmatrix} 1 & 0 & 1 \\\0 & 1 & 0 \\\0 & 0 & 1 \end{pmatrix}$ in $GL_3(\mathbb R)$. Its centralizer is $\Big\\{ \begin{pmatrix} a & b & c \\\ 0 & d & e \\\ 0 & 0 & a \end{pmatrix} \Big\\} \subset GL_3(\mathbb R)$. Although this is a solvable group, and looks very similar to the Borel in $GL_3$, which is not unimodular, it is in fact a unimodular group (as the OP points out in a second comment below).