Of course! It appears 'import' is a special case, presumably because many modules don't define it, yet it's always called by 'use'.

C:\>perl -w -MStrict -le "Strict->import"

C:\>perl -w -MStrict -le "Strict->foo"
Can't locate object method "foo" via package "Strict" 
(perhaps you forgot to load "Strict"?) at -e line 1.
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So how was the OP getting a subroutine redefined error in the first place if the import wasn't getting called?

FooBar.pm

package FooBar;
require Exporter;
@ISA = qw(Exporter);

@EXPORT_OK = qw(foo);

sub foo {
  print "foo called\n";
}

1;
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test1.pl

#!/usr/bin/perl
use strict;
use warnings;
use FooBar qw(foo);

foo();
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> foo called

test2.pl

#!/usr/bin/perl
use strict;
use warnings;
use foobar qw(foo);

foo();
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> Undefined subroutine &main::foo called at foo.pl line 7.

This suggests to me that the File::Basename->import would have worked, but the File::BaseName->import would not have worked, thus redefining nothing.