use strict;
use warnings;
use 5.010;
 
my $str = <<"ENDOFDATA";
VECT
1 1 1
2 2 2
3 3 3
 
VECT
4 4 4
5 5 5
6 6 6
ENDOFDATA
 
#open() your original file here:
open my $INFILE, '<', \$str
    or die "Couldn't read from string: $!"; 

 
my $section_counter = 1;
 
{
  local $/ = ""; #tell perl that the end of a line will
                 #be reached when perl encounters any
                 #number of consecutive blank lines

  while (my $section = <$INFILE>){
      chomp($section);

      my $fname = "file$section_counter";
      say $fname;

      say $section;  #or write to a file
      say '-' x 20;

      $section_counter++;
  }
}
 
close $INFILE;

--output:--
file1
VECT
1 1 1
2 2 2
3 3 3
--------------------
file2
VECT
4 4 4
5 5 5
6 6 6
--------------------