The answer is that the FPP is not absolute, and indeed, even the unit interval loses the FPP in a forcing extension. The unit interval famously has the FPP, but I claim that in any forcing extension having a new real, such as the forcing extension $V[c]$ obtained by adding a Cohen real, which preserves cardinals, the ground model unit interval no longer has the FPP.
To see this, let $X=I^V$ be the unit interval of $V$, considered as a topological space in $V[c]$. Let $a_n\to c$ be an increasing sequence of rational numbers converging to $c$ from below, and $b_n\to c$ from above. Let $f:X\to X$ be the piece-wise linear function that takes the interval $[a_n,a_{n+1}]\to [a_{n+1},a_{n+2}]$ and similarly $[b_{n+1},b_n]\to [b_{n+2},b_{n+1}]$. This function takes ground-model reals to ground-model reals, but below $c$, it lies above the diagonal and above $c$, it lies below the diagonal. So it has no fixed point in $X$. (Meanwhile, it has a natural extension to the unit interval in $V[c]$, having fixed point $c$.)