  This paper brings new results on the FPP in Banach spaces $X$ with a Schauder
basis. We first deal with the problem of whether there is a Banach space
isomorphic to $\co$ having the FPP. We show that the answer is negative if $X$
contains a pre-monotone basic sequence equivalent to the unit basis of $\co$.
We then study sufficient conditions to ensure the existence of such sequences.
Interesting results are obtained, including the case when $X$ has a
$1$-suppression unconditional basis and its unit ball fails the PCP. With
regarding the weak-FPP, we establish two fixed-point results. First, we show
that under certain conditions this property is invariant under Banach-Mazur
distance one. Next, it is shown that when the basis is either $1$-suppression
unconditional or $1$-spreading then $X$ has the weak-FPP provided that a
Rosenthal's type property on block basis is verified.
