Your method of getting the next digit is expensive! Replace

  for my $digit (@digits) {
    next if $digit ~~ @number;
    push @number, $digit;
    append_digit();
    pop @number;
  }
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with

  for my $pos (0..$#digits) {
    push @number, $digits[$pos];
    local @digits = @digits[0..$pos-1, $pos+1..$#digits];
    append_digit();
    pop @number;
  }
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to cut down the time taken to 2/3 (6s from 9s).

Also, there's reason to use a hash (outside of testing).

Full code:

#!perl
use strict;
use warnings;
use 5.010;

our @digits = 1..9;

my @number;
my @results;
sub append_digit {
  my $number = join "", @number;
  
  push @results, $number
     if @number && !grep $number % $_, @number;

  for my $pos (0..$#digits) {
    push @number, $digits[$pos];
    local @digits = @digits[0..$pos-1, $pos+1..$#digits];
    append_digit();
    pop @number;
  }
}

my $stime = time;
append_digit();
my $etime = time;
say $etime-$stime;

say "count: " . (0+@results);
#say for sort { $a <=> $b } @results;
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