Solved it:

Copyapt-get install libzip-dev

will install the missing libzip-distribution

To install libzip4, you have to add the Ubuntu repositories to your apt source list. This can be done by

sudo apt-add-repository main
sudo apt-add-repository restricted
sudo apt-add-repository universe
sudo apt-add-repository multiverse

Here you find some examples.

Ok, so the issue is that you have to guess that the package you need to install is called "libzip-dev", not just "libzip".

So the solution was :

RUN apt-get update \
     && apt-get install -y libzip-dev \
     && docker-php-ext-install zip

I have managed to compile PECL zip (partly due to snow in the UK cancelling my booked day!). I have not tested this in php, but it loads correctly into phpinfo();

Option 1: Update Libzip & Hope for the best

I used an external CentOS-7 repo (which amzn linux 2 is supposed to be pretty compatible with) and the following script:

#PECL requires libzip 11 - get it from a foreign repo
sudo rpm --import http://wiki.psychotic.ninja/RPM-GPG-KEY-psychotic
#this looks wrong but the repo uses a single release
sudo rpm -ivh http://packages.psychotic.ninja/6/base/i386/RPMS/psychotic-release-1.0.0-1.el6.psychotic.noarch.rpm

#YUM REPO HACKS START
#unfortunately amazon-linux-2 release numbering breaks it - force RHEL 7 numbering
sudo sed -i s/\$releasever/7/g /etc/yum.repos.d/psychotic.repo
#Need to exclude outdated amzn zip packages
sudo yum remove libzip libzip-devel
#check for an exclude line, add our one if not present (to the main repo)
grep -A 100 -m1 "\[amzn2-core" /etc/yum.repos.d/amzn2-core.repo | grep exclude= || sudo sed -i -E "s/($(grep -A 100 -m1 "\[amzn2-core" /etc/yum.repos.d/amzn2-core.repo | grep -m1 ^name=)$)/\1\nexclude=libzip libzip-devel/" /etc/yum.repos.d/amzn2-core.repo
#check this worked, if not modify the existing exclude line(s)
grep -A 100 -m1 "\[amzn2-core" /etc/yum.repos.d/amzn2-core.repo | grep "exclude=.*libzip" || sudo sed -i -E "s/exclude=/exclude=libzip libzip-devel /" /etc/yum.repos.d/amzn2-core.repo
#YUM REPO HACKS END

sudo yum install --enablerepo=psychotic-plus gcc libzip libzip-devel
#now install PECL
wget http://pear.php.net/go-pear.phar
sudo php go-pear.phar
#patch it to cope with the XML being a module
sudo sed -i "$ s|\-n||g" /usr/bin/pecl
#and this should now work
sudo pecl install zip
#may still need to manually add to the php.ini
grep "^extension=zip.so" /etc/php.ini || echo "extension=zip.so" | sudo tee -a /etc/php.ini

Be VERY careful with the bit of this script marked 'YUM REPO HACKS' - it intentionally includes commands to modify your repo configuration which have had minimal testing. All it is actually doing is replacing the '$releasever' variable in the pyschotic repo with '7' and adding 'exclude=libzip libzip-devel' to the amzn2-core repo. You could do this with vim!

Option 2: 2 versions of libzip

From my research: if you are not comfortable with a 'found-by-google' repo on your server (I don't blame you, depends on use case) then you could leave the amzn zip in place, use their source repo to install libzip 11.2 source, install that (and/or its libraries) to an alternate location (find the instructions elsewhere for that) and compile the php module manually.

To compile zip manually:

$ pecl download zip
$ tar -xf zip-*.tgz && cd zip-* && phpize
$ ./configure --with-libzip=/path/to/secondary/libzip

...etc

You can install it from the Ubuntu 20.04 repositories.

wget http://mirrors.kernel.org/ubuntu/pool/universe/libz/libzip/libzip5_1.5.1-0ubuntu1_amd64.deb
sudo apt install ./libzip5_1.5.1-0ubuntu1_amd64.deb