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Caution: This needs an overhaul to reflect USB stick layout changes (ESP partition). Could not reproduce it working with a debian-live-11.2-amd64 image.

The next step will allow you to get a debian live usb with persistence:

Get the iso

Download an iso-hybrid of debian: https://www.debian.org/CD/live/. Copy the iso image to your empty USB stick. All data on it will probably be erased. In the examples below, /dev/sdd is assumed to be the USB stick device. You've got to check your device with e.g. dmesg|tail.

Create additional partitions on the drive

We can use:

Prepare the files

Lets copy files to newly created partitions.

Changes required to existing FAT partition

We have to mount the FAT partition, lets mount /dev/sdd2 to /mnt. Then unarchive the iso to /mnt. 

# mount /dev/sdd2 /mnt
# cd /mnt
# 7z x /home/user/iso/debian-live.iso

??? We need to do some changes on the files. 

# mv isolinux syslinux
# for f in $( grep -FRil "splash quiet" ); do echo $f; LANG=C sed 's/splash quiet/persistence /;s/quiet splash/persistence /' -i $f; done;
# cd syslinux
# rename "s/iso/sys/" iso*
# cd ~
# sync
# umount /dev/sdd1

EXT4 partition with label Persistence

Create a new ext4 partition with label persistence. The label "persistence" is mandatory.

We need to put a conf on the persistence partition. 

# mount /dev/sdd3 /mnt
# cd /mnt
# echo "/ union" > persistence.conf 
# sync
# umount /dev/sdd3