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This paper brings new results on the FPP in Banach spaces $X$ with a Schauderbasis. We first deal with the problem of whether there is a Banach spaceisomorphic 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. Withregarding the weak-FPP, we establish two fixed-point results. First, we showthat under certain conditions this property is invariant under Banach-Mazurdistance one. Next, it is shown that when the basis is either $1$-suppressionunconditional or $1$-spreading then $X$ has the weak-FPP provided that aRosenthal's type property on block basis is verified.

Remarks on the FPP in Banach spaces with unconditional Schauder basis