Answering my own question, I guess: It works if I change mirrors.

myuser@spangled:~$ cpanm -M https://cpan.metacpan.org HTML::Template
--> Working on HTML::Template
Fetching https://cpan.metacpan.org/authors/id/S/SA/SAMTREGAR/HTML-Temp
+late-2.97.tar.gz ... OK
Configuring HTML-Template-2.97 ... OK
==> Found dependencies: CGI
--> Working on CGI
Fetching https://cpan.metacpan.org/authors/id/L/LE/LEEJO/CGI-4.44.tar.
+gz ... OK
Configuring CGI-4.44 ... OK
==> Found dependencies: Test::Warn, Test::NoWarnings, Test::Deep, HTML
+::Entities
--> Working on Test::Warn
Fetching https://cpan.metacpan.org/authors/id/B/BI/BIGJ/Test-Warn-0.36
+.tar.gz ... OK
Configuring Test-Warn-0.36 ... OK
==> Found dependencies: Sub::Uplevel
--> Working on Sub::Uplevel
Fetching https://cpan.metacpan.org/authors/id/D/DA/DAGOLDEN/Sub-Upleve
+l-0.2800.tar.gz ... OK
Configuring Sub-Uplevel-0.2800 ... OK
Building and testing Sub-Uplevel-0.2800 ... OK
Successfully installed Sub-Uplevel-0.2800
Building and testing Test-Warn-0.36 ... OK
Successfully installed Test-Warn-0.36
--> Working on Test::NoWarnings
Fetching https://cpan.metacpan.org/authors/id/A/AD/ADAMK/Test-NoWarnin
+gs-1.04.tar.gz ... OK
Configuring Test-NoWarnings-1.04 ... OK
Building and testing Test-NoWarnings-1.04 ... OK
Successfully installed Test-NoWarnings-1.04
--> Working on Test::Deep
Fetching https://cpan.metacpan.org/authors/id/R/RJ/RJBS/Test-Deep-1.12
+8.tar.gz ... OK
Configuring Test-Deep-1.128 ... OK
Building and testing Test-Deep-1.128 ... OK
Successfully installed Test-Deep-1.128
--> Working on HTML::Entities
Fetching https://cpan.metacpan.org/authors/id/G/GA/GAAS/HTML-Parser-3.
+72.tar.gz ... OK
Configuring HTML-Parser-3.72 ... OK
==> Found dependencies: HTML::Tagset
--> Working on HTML::Tagset
Fetching https://cpan.metacpan.org/authors/id/P/PE/PETDANCE/HTML-Tagse
+t-3.20.tar.gz ... OK
Configuring HTML-Tagset-3.20 ... OK
Building and testing HTML-Tagset-3.20 ... OK
Successfully installed HTML-Tagset-3.20
Building and testing HTML-Parser-3.72 ... OK
Successfully installed HTML-Parser-3.72
Building and testing CGI-4.44 ... OK
Successfully installed CGI-4.44
Building and testing HTML-Template-2.97 ... OK
Successfully installed HTML-Template-2.97
8 distributions installed
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