In each iteration of your foreach loop, $_ is set to one value in @files. As your call to Tr is inside the foreach loop, $_ has the same value in each td. You can use map to iterate over the list of files and get the results you want. Also, note the use of File::Basename::basename instead of a regular expression to remove the directory from the filenames:

use File::Basename;
# ...
# 4 elements at a time
while( my @cols= splice @files, 0, 4 ) {
  print Tr( {-align=>'center'},
    map {
      td( $_ ? submit(-name=>'file', -value=> basename($_)) : ' '
+ )
    } @cols[0..3]
  );
}
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Or, use the distributive properties of the HTML shortcuts provided by CGI.pm (see the pod). In this case, we are passing a reference to a list of values to td, rather creating a list of calls to td:

use File::Basename;
# ...
while( my @cols= splice @files, 0, 4 ) {
  print Tr( {-align=>'center'},
    td([
      map {
        $_ ? submit(-name=>'file', -value=> basename($_)) : ' '
      } @cols[0..3]
    ])
  );
}
[download]