Went to Apple Store in Perth (Australia) after being without a proper phone for four weeks.
They replaced the display of the phone which ultimately fixed the issue. The phone was 1.5 years old - the cost of the screen replacement was covered under Australian consumer law.
It appears that only a small number of users are impacted by this issue - all after upgrading to iOS 11.2.5.
Since replacing the display seems to fix the issue, I assume that it is being caused by iOS 11.2.5 in combination with a specific version (or batch) of the touch screen.
Update to 11.2.5 on my second iPhone SE worked without any issues.