The problem now is I am using timestamp in seconds and there could be more than one name generated in the same second with same user provided name.

How about using a simple incrementing number in this case? You can use sysopen to do this and avoid race conditions:

use Fcntl;

my $username = "...";
my $filename = $username . time;
my $inc = 1;

my $fh;

until (sysopen $fh, $filename.$inc, O_WRONLY|O_CREAT|O_EXCL) {
  $inc++;
}

print $fh time, "\n";
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