It passes all your test cases :)

#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11130527
use warnings;

local $_ = <<END;
/usr/vsa/pkgs/python3/3.6.3a/bin/python3.6
/usr/vsa -> /root/site/tools/gauv
/usr/vsa/pkgs/python3/3.6.3a -> 3.6.3
END

s!^(\S+/)([^\s/]+) -> \K([^/]\S*)!$1$3!gm; # fixup relative
my $was = '';
while( $was ne $_ )
  {
  my %links;
  $links{$1} = $2 while /^(\S+) -> (\S+)/gm;
  $was = $_;
  for my $link ( sort keys %links )
    {
    s~(?<!\S)\Q$link\E(?=/)~$links{$link}~; # replace links
    }
  }

my %subpaths;
m~^(\S+)(?=[\s/])(?{$subpaths{$1}++})(*FAIL)~m; # get all subpaths
my @subpaths = sort { length $a <=> length $b } sort keys %subpaths;
print "$_\n" for @subpaths;
[download]

Outputs:

/usr
/root
/usr/vsa
/root/site
/root/site/tools
/root/site/tools/gauv
/root/site/tools/gauv/pkgs
/root/site/tools/gauv/pkgs/python3
/root/site/tools/gauv/pkgs/python3/3.6.3
/root/site/tools/gauv/pkgs/python3/3.6.3a
/root/site/tools/gauv/pkgs/python3/3.6.3/bin
/root/site/tools/gauv/pkgs/python3/3.6.3/bin/python3.6
[download]