Revive without loosing data?

My S4 mini went dead (no boot logo, no vibration, screen just stayed dark when trying to switch it on). During normal usage, it froze and after disconnecting the battery, it was stonedead (official Samsung Android 4.4.2 image, not rooted).

It seems, the eMMC was damaged.
Searing "xda forums" ;) I tried a "debrick.img" via SD card boot- but only got "emmc read error".

As last idea I tried "heating" up the eMMC using a SMD soldering station from a friend (trying to "heal" cold soldering joints, which might be caused by thermal and/or physical stress) - and it worked somehow: I got a boot logo, download mode and recovery mode. But still in a bootloop...


So my question is:
Is there any method to restore user data from the flash?
Or flash without destroying user data?
Recovery is still the original samsung...
I guess, flashing a new image via ODIN, would kill all my data...

I'm also curious., if there is any image completely booting from SD card (so I can manually mount the flash partitions).

Any hint is appreciated. Thanks for help in advance!


PS:
Thanks to all for the great information here!
I was really asthonished, that the CPU is trying to boot via SD card...

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