This appears to meet your spec:

#!/usr/bin/perl
use strict;
use warnings;
use 5.10.0;
use Data::Dumper;

my $beginString = "SEARCH";
my $endString = "TEST";


my $fileContent;
{
    local $/;
    $fileContent = <DATA>;
}



if($fileContent =~ /\b$beginString\b(.*?)\b$endString\b/s){
    my $result = $1;
    print $result;
}

#print Dumper($fileContent); 

__DATA__
#Ignoring this
SEARCH => 
#Beginning of data I need
.......

........

 #all the data I need
.......


.....
#End of Data I Need
TEST =>

# And ignoring this
[download]

Given that it's largely taken from your posted code, I wonder if you accidentally fixed your issue in preparing the post. If this does not output what you need, please describe how.

Sidenote 1:

my $fileContent;
open(my $fileHandler, $inputFile) or die "Could not open file '$inputF
+ile' $!";
{
    local $/;
    $fileContent = <$fileHandler>;
}
close($fileHandler);
[download]

You can rewrite this as

my $fileContent = do {
    open(my $fileHandler, $inputFile) or die "Could not open file '$in
+putFile' $!";
    local $/;
    <$fileHandler>;
};
[download]

and avoid scoping challenges on that file handle.

Sidenote 2: When using start/finish tags, assuming you want literals, you should quotemeta:

if($fileContent =~ /\b\Q$beginString\E\b(.*?)\b\Q$endString\E\b/s){
[download]

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