There are a couple of issues with your code

In your first while loop, you have a bareword sp where you meant $sp. The effect of this would be to assign only one value to each key in the file. Second and subsequent values for a key would be assigned to $p2i{sp}.

In your first foreach loop you try to print the value of $p2i{$key} which is simply a reference to the array you have stored the values in. You have to dereference this value into an array with @{$p2i{$key}}

You have some duplicative code in your if(exists...) loop. You can print the values of the array in $p2i{$user} with the simple foreach loop.

For more information on references, take a look at perldoc -m perlreftut and perldoc -m perlref

while(<FILE>) {
  chomp;
  ($sp, $ipr) = split;
  if (exists $p2i{$sp}) {
    push @{$p2i{$sp}}, $ipr;
#    push @{$p2i{sp}}, $ipr;
  }
  else {
    $p2i{$sp}= [$ipr];
  }
}

close FILE;

# test to see if hash created correctly

foreach $key (sort keys %p2i) {
#  print "$key\t$p2i{$key}\n";
  print "$key\t@{$p2i{$key}}\n";
}

if (exists $p2i{$user}) {
#        print "$user is in the hash. \n";if (exists $p2i{$user}) {
    print "$user is in the hash. \n";
#    @user_array = split(/ /, $p2i{$user});
#    @user_array = split(/ /, $p2i{$user});
#    foreach (@user_array) {
#      print "$_\n";
#    }
    foreach (@{$p2i{$user}}){
      print "$_ ";
    }
    print "\n";
  # print "$user: @{ $p2i{$user} }\n";  
  # only retieves one value even when more than one   #val even if the
+re are multiple
#  }
}
else {
  print "$user is not in the hash.\n";
}

exit;
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