My previous timings where just from a single run of each program. Using Benchmark and repeating to verify consistent results, I find that foreach is indeed faster.

use strict;
use warnings;
use Benchmark;

my @a;
$a[$_] = int( rand(100) ) for ( 0 .. 10000000 );

cmpthese(
    -30,
    {
        'foreach'      => sub { foreach (@a) { $_ += 1; } },
        'map (assign)' => sub { @a = map { $_ += 1 } @a; },
        'map (bare)'   => sub { map { $_ += 1 } @a; },
    }
);
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Results:

             s/iter map (assign)   map (bare)      foreach
map (assign)   2.06           --         -40%         -76%
map (bare)     1.23          67%           --         -59%
foreach       0.501         311%         146%           --
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