You need to pass an array reference to the -values parameter, not the whole array itself. Adding labels to change what the user sees, you get something like this:

use CGI qw(:all);

my @files = <../data/texts/*.txt>;
my %labels = (
    map { $_ => +( m!([^/]+)\z! ) } @files
);

print popup_menu(
    -name   => 'select',
    -values => \@files,
    -labels => \%labels
);

__END__
<select name="select">
<option value="../data/texts/foo.txt">foo.txt</option>
<option value="../data/texts/bar.txt">bar.txt</option>
</select>
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