I have read that the 64-bit Strawberry Perl includes the "gcc" compiler but the 32-bit Strawberry Perl doesn't include "gcc".

I don't know where you read that, but it's verifiably false: I just downloaded http://strawberryperl.com/download/5.24.1.1/strawberry-perl-5.24.1.1-32bit-portable.zip, and on inspecting the contents of the zip file, I see the gcc compiler zip:\c\bin\gcc.exe and related files.

I've also never seen any of my strawberry installations try to use gmake instead of the dmake that comes with strawberry. Did you change the configuration of cpanm?

I haven't installed the 5.24.1.1-32, but my 5.24.0.1-32 installation doesn't seem to complain about gmake or makefile.gcc. You might have something getting in the way of your path to the gcc compiler or dmake or something. From a cmd.exe window, run

perl -v
where perl
where gcc
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C:\Users\pryrt\Documents>path c:\usr\local\apps\berrybrew\perls\5.24.0
+_32\c\bin;c:\usr\local\apps\berrybrew\perls\5.24.0_32\perl\bin;c:\usr
+\local\apps\berrybrew\perls\5.24.0_
32\perl\site\bin;%PATH%

C:\Users\pryrt\Documents>perl -v

This is perl 5, version 24, subversion 0 (v5.24.0) built for MSWin32-x
+86-multi-thread-64int

...

C:\Users\pryrt\Documents>where perl
c:\usr\local\apps\berrybrew\perls\5.24.0_32\perl\bin\perl.exe
C:\usr\local\apps\strawberry\perl\bin\perl.exe

C:\Users\pryrt\Documents>where gcc
c:\usr\local\apps\berrybrew\perls\5.24.0_32\c\bin\gcc.exe
C:\usr\local\apps\strawberry\c\bin\gcc.exe
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With that, when I run cpanm Alien::wxWidgets, I get

C:\usr\local>cpanm Alien::wxWidgets
WARNING: Module::Signature and Digest::SHA is required for distributio
+n verifications.
--> Working on Alien::wxWidgets
Fetching http://www.cpan.org/authors/id/M/MD/MDOOTSON/Alien-wxWidgets-
+0.67.tar.gz ... OK
Configuring Alien-wxWidgets-0.67 ... OK
Building and testing Alien-wxWidgets-0.67 ...
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I was able to write this whole post in the time it takes to build and test, and am having to wait before posting, but I haven't seen the gmake errors you mentioned in the first 5 minutes -- after another 5 minutes, still going... but looking at %HOME%\.cpanm\work\...\build.log, I see the log updating, and there have been hundreds of lines of successful instances of gcc and g++. If it had a "gmake" problem, I would think it would have shown up before doing any compiling. Thus, I will assume it won't, and create my post. If I'm wrong, I will edit my post.

To reiterate, I think you have a PATH issue, or have a faulty installation of strawberry.

update: 15 min later:

C:\usr\local>cpanm Alien::wxWidgets
WARNING: Module::Signature and Digest::SHA is required for distributio
+n verifications.
--> Working on Alien::wxWidgets
Fetching http://www.cpan.org/authors/id/M/MD/MDOOTSON/Alien-wxWidgets-
+0.67.tar.gz ... OK
Configuring Alien-wxWidgets-0.67 ... OK
Building and testing Alien-wxWidgets-0.67 ... OK
Successfully installed Alien-wxWidgets-0.67
1 distribution installed
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You also might want to check for

where dmake
where cpanm
[download]

update Mar 10: I learned (was reminded?) today that strawberry comes with both dmake and gmake, so I will strike my comments about never seeing strawberry use gmake... I had just not noticed (or remembered?) that I'd seen it. :-)

Strawberry has always shipped with gcc, associated build tools, non core modules and libraries. The point is to provide a complete development environment on Windows. This sounds more like an installation or configuration problems.