This is untested but looks reasonable.

#!/usr/bin/perl -w

use strict;

my $delimiter = qr/================================/;
my $counter   = 0;

# If the first line of every input file is a delimiter, you
# can comment this out and the first output file will be
# opened on the first iteration of the while() loop.
open FH, ">file$counter"
    die "could not open file$counter: $!";

# Read each file in @ARGV or stdin one line at a time
while (<>)
{
    if (/$delimiter/)
    {
        # If we hit a delimiter, close the currently
        # open file, and open a new one.
        $counter++;
        close FH;
        open FH, ">file$counter"
            or die "could not open file$counter: $!";
    }  else {
        # Any line other than a delimiter goes into
        # the currently open file
        print FH;
    }
}
[download]

-Matt