You can keep these first words in a hash and check if they have already been stored:

#!/usr/bin/perl
use strict;
use warnings;

my %seen_words;

while (<DATA>){
    if (!m/^(\S+)/){
        die "Invalid line: $_";
    } 
    my $first_word = $1;
    if (!$seen_words{$first_word}){
        print;
        $seen_words{$first_word} = 1;
    }

}

__DATA__
MA01001A1A03.f1 760 5640111 ad1
MA01001A1A03.f1 760 42572233 ubq
MA01001A1A04.f1 300 15232924 ubq
MA01001A1A04.f1 300 145334669 DNA
MA01001A1B22.f1 580 77745475 ra
MA01001A1B22.f1 580 30409730 ra
[download]

This can be written a little bit compacter:

while (<DATA>){
    if (!m/^(\S+)/){
        die "Invalid line: $_";
    } 
    print unless $seen_words{$1}++;
}
[download]

But the first one is easier to read for the beginner ;-)