Thanks for this quick reply!

Unfortunately I don't have a desktop PC to test the card. My chassis is this one: https://www.owc.com/solutions/mercury-helios-3s

No driver needed, certified for windows and Mac, no word about Linux.
Test the interface on another computer without this chassis, otherwise you never know if the issue is from the chassis or not.

Well, my machine's driver is snd_hdsp which (I guess) came with the aur/snd-hdspe-dkms package (i think)

$ lsmod | grep snd_hdsp
snd_hdsp               94208  1
snd_rawmidi            53248  4 snd_seq_midi,snd_usbmidi_lib,snd_hdsp,snd_ump
snd_hwdep              20480  3 snd_usb_audio,snd_hda_codec,snd_hdsp
snd_pcm               200704  15 snd_soc_avs,snd_hda_codec_hdmi,snd_hda_intel,snd_usb_audio,snd_hda_codec,soundwire_intel,snd_sof,snd_hdsp,snd_sof_intel_hda_common,snd_compress,snd_soc_core,snd_sof_utils,snd_soc_skl,snd_hda_core,snd_pcm_dmaengine
snd                   155648  27 snd_ctl_led,snd_hda_codec_generic,snd_seq,snd_seq_device,snd_hda_codec_hdmi,snd_hwdep,snd_hda_intel,snd_usb_audio,snd_usbmidi_lib,snd_hda_codec,snd_hda_codec_realtek,snd_sof,snd_hdsp,snd_timer,snd_compress,thinkpad_acpi,snd_soc_core,snd_ump,snd_pcm,snd_rawmidi

.
There is another package snd-hdspe-dkms-git which is the development branch of the driver's repository. I've been hesitant to install that but maybe I should give it a try...

I can't say anything about the chassis' firmware - there's no mention on that on the manufacturer's website, nor any download link for updates. I've read on other manufacturer's websites (e.g. Sonnet) about Linux compliance but when it came to RME PCIe card compliance they also only spoke about Windows/macOS but not Linux compliance (they're also a bit more expensive).

What else?

$ cat /proc/asound/card0/hdsp 
RME Hammerfall DSP + Multiface (Card #1)
Buffers: capture 0000000010e4ec40 playback 0000000039c9cc81
IRQ: 18 Registers bus: 0xc4000000 VM: 0xffffc9000104a000
Control register: 0x10040d8
Control2 register: 0x0
Status register: 0x2000000
Status2 register: 0xc0010703
FIFO status: 0
MIDI1 Output status: 0x33fac100
MIDI1 Input status: 0x33fac100
MIDI2 Output status: 0x300
MIDI2 Input status: 0x300
Use Midi Tasklet: on

Buffer Size (Latency): 1024 samples (2 periods of 4096 bytes)
Hardware pointer (frames): 0
Precise pointer: off
Line out: on
Firmware version: 5

Sample Clock Source: Internal 48 kHz
Preferred Sync Reference: ADAT1
AutoSync Reference: None
AutoSync Frequency: 0
System Clock Mode: Master
System Clock Frequency: 48000
System Clock Locked: Yes

IEC958 input: Coaxial
IEC958 output: Coaxial only
IEC958 quality: Consumer
IEC958 emphasis: off
IEC958 NonAudio: off
IEC958 sample rate: Error flag set

ADAT1: No Lock
SPDIF: No Lock
Word Clock: No Lock
ADAT Sync: No Lock

... this pretty much reflects what alsamixer is showing me. Anything to be said about that?

Thanks!

Thanks very much for all input on my question! I've meanwhile tried a couple things but no luck so far

I've tried to flash the PCIe card with a newer firmware from my Windows partition (I think there's a revision 21 in fut_wdm_dspe.zip which I downloaded from the RME website). However, the soundcard then totally disappeared, so, I reverted that (luckily the installer let me install a legacy firmware which, I think, is revision 20)
As the chassis is using the xhci_hcd kernel driver I recognized that during kernel compilation there was some note about missing firmware for xhci_hcd. I installed the firmware but that didn't change anything.

Well, at least I can listen to some music, watch some videos or even user JACK if I set the buffersize to 1048 - not ideal but better than nothing at all.

Maybe just a naive question: Could there be a difference between a Multiface I and a Multiface II?