Finally, I got it working. I took care about the gdb server. A JLinkGDBServerCLExe existed from an installed STM32 IceCube (experimented two days ago), but that seemed not to work (I don't remember what was wrong). Today I installed Segger's JLink_Linux .deb-package and I reprogrammed DAPLink with the Segger firmware. Install of the .deb-package initially resulted in an error: the installer failed to overwrite a /etc/udev/99-jlink.rules (quite sure from former IceCube install). Even renaming did not work. Only after executing sudo dpkg --remove segger-jlink-udev-rules, install of segger .deb-package succeeded.
Testing with the NXP MCUXpresso example dad_adc now did it! Automatically configured via GDB SEGGER Interface Debugging. Then I tried the Zephyr application. Duplicating the configuration for the former project did not work (I spend some words towards the end of this posting). So I went to follow the instructions of the (already) mentioned pdf "Build Your First Zephyr Application on i.MX RT".
Finally, it works using the GDB SEGGER J-Link Debugging scheme. In contrast to there, I had to fill in another device name (MK64FN1M0xxx12) and also put an absolute executable path (/usr/bin/JLinkGDBServerCLExe). The same applies to the GDB client setup (here: /usr/local/mcuxpressoide-11.5.1_7266/ide/plugins/com.nxp.mcuxpresso.tools.linux_11.5.1.202201181444/tools/bin/arm-none-eabi-gdb). I had to set the usual macro variables for Zephyr project import by hand in order to be able to fill in a plain arm-none-eabi-gdb instead of the full path.