Actually I was interested in what kinds of trouble you ran into, but anyway...

There's a trick in Perl that makes that quite easy: you can set the input record separator to '>', so that the <DATA> iterator gives you chunks splitted by '>':

use strict;
use warnings;

local $/ = '>';

my $c = 0;
while (<DATA>) {
    chomp;
    next unless length;
    my $fn = "file" . ++$c;
    open my $handle, '>', $fn or die "Can't open `$fn' for writing: $!
+";
    print $handle '>', $_;
    close $handle or warn $!;
}

__DATA__
>i:/13414 Ccl9 (88)/sequencing/13414_fasta.Contig1
TTTCTCCGGCCCCCTCCTCCCGCGGGGGAAAAAACCCGGGGAGCAGTCGG
GCAGGGGTTTTTTGGTTTTTTCAAAATAAAAAGGGGTGCCCGTTGGGGGGcd 
GGGGGGGTGCAGGTTTCAACCCCCCCCCCCAAAGAAAAAAAAATTTTGGG
GAATTTTTGGGGGGCTCCACCAGTTTTCGGGGTTTTTGGGCCTTTTCAGA
AGGTAGGTTGGACGCGGATTGGGCAATAAACCACCCCGCTTCATCGGATA
ATTTTCCCCGGCCGAAAAGGGCCGCGGGGCCGGTGGGCGGCCTTGGGTTT
>i:/13414 Ccl9 (88)/sequencing/13414_fasta.Contig2
TAAACCCAAGGCCCCCCAGGTAAAAAAAAAACCGGCCAGGGGGGGGGGGG
TAAAAAAAACCAAGTGTCACCCAGGGTGGAGATCCCCGGAAAAGGAAAAG
GGGGGTTTTTTATTCGAAACGGGGAAAACTTTCACAAAATTTTGGAAGAA
TCCCCTTTAATGTTTTCTTTTCAAAAGGGGGTAAAAAAACCACCTTTAAA
AAGAAGTCTACCTTGGGAAAAAATAATTTTTGGGAAAATTTAAAAATTGA
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