It works if you add an end marker for the right part:

use Data::Dump qw( pp );

my $str = <<_BLA_;
SECTION 1
bla bla
bla
SECTION 2
blabla
blabla
SECTION 3
bla
_BLA_

my %hash = $str =~ /(SECTION \d+)\s*(.*?)(?=SECTION|$)/sg;
pp \%hash;

__DATA__
{
  "SECTION 1" => "bla bla\nbla\n",
  "SECTION 2" => "blabla\nblabla\n",
  "SECTION 3" => "bla",
}
[download]

Edit: \s* rather than \s+ to allow empty section at EOF