There's no reason why that should be. Why would <> returns something different based on the input's extension.

>perl -MO=Deparse -pe"}{$\=$."
LINE: while (defined($_ = <ARGV>)) {
    ();
}
{
    $\ = $.;
}
continue {
    die "-p destination: $!\n" unless print $_;
}
-e syntax OK
[download]

And I don't see it happening

>perl -pe"}{$\=$." script.pl
56
>copy script.pl test.txt
        1 file(s) copied.

>perl -pe"}{$\=$." test.txt
56
[download]

Provide a hexdump of your text file if you want to look into this further.