  In this paper we establish some new results concerning the Cauchy-Peano
problem in Banach spaces. Firstly, we prove that if a Banach space $E$ admits a
fundamental biorthogonal system, then there exists a continuous vector field
$f\colon E\to E$ such that the autonomous differential equation $u'=f(u)$ has
no solutions at any time. The proof relies on a key result asserting that every
infinite-dimensional Fr\'echet space with a fundamental biorthogonal system
possesses a nontrivial separable quotient. The later, is the byproduct of a
mixture of known results on barrelledness and two fundamental results of Banach
space theory (namely, a result of Pe{\l}czy\'nski on Banach spaces containing
$L_1(\mu)$ and the $\ell_1$-theorem of Rosenthal). Next, we introduce a natural
notion of weak-approximate solutions for the non-autonomous Cauchy-Peano
problem in Banach spaces, and prove that a necessary and sufficient condition
for the existence of such an approximation is the absence of $\ell_1$-isomorphs
inside the underline space. We also study a kind of algebraic genericity for
the Cauchy-Peano problem in spaces $E$ having complemented subspaces with
unconditional Schauder basis. It is proved that if $\mathscr{K}(E)$ denotes the
family of all continuous vector fields $f\colon E\to E$ for which $u'=f(u)$ has
no solutions at any time, then $\mathscr{K}(E)\bigcup \{0\}$ is spaceable in
sense that it contains a closed infinite dimensional subspace of $C(E)$, the
locally convex space of all continuous vector fields on $E$ with the linear
topology of uniform convergence on bounded sets.
