Are there pancake lenses for the A7C? While in Japan I saw a lot of older japanese guys with some camera which I couldn't identify, it looked like some digital remake of some older camera, but what struck me was that they all had this pancake lens which made the whole package very thin and small.
My A7C is also very compact already but the 35mm 1.8 lens is, while very compact, by far not flat, so I can't put it in my pocket for example.
It doesn't need to be very fast, because I will use it mostly outside in good light.
Regarding “digital remake of some older camera”: there are a lot of APS-C and MFT cameras that resemble older cameras, like the Nikon Zfc, which will have smaller lenses because of the smaller sensors. A good example is Sonys 20mm F2.8 which is super small but only APS-C compatible.
I had the Sony Zeiss 35mm f2.8 and for FF I think it was pretty compact. You could also look into manual lenses, for example the 50mm f2 from TTArtisan is very small and affordable.
A7C is E-mount right? There’s the 16mm F2.8, 20mm F2.8, and PZ 16-50mm F3.5-5.6 OSS that are all more compact than the 35mm F1.8. They’re cheap lenses too.
Something to keep in mind: the lenses you have listet are alle only APS-C compatible and therefore would go into crop mode on an A7C. You can use the lenses on the A7C but it would beg the question: why buy a FF camera to only use it in crop mode?
As others have hit on already, sensor size is the thing holding you back. Bigger image circles require more glass to resolve. The most compact FF e-mount option I'm aware of is Zony 35mm f2.8
Here's a quick size rundown. The zFc is very classic looking, so I included it. Had a n APS-C and is wearing a FF lens though. Next up is your A7c with the previously mentioned lens, followed by a fairly compact micro four thirds body and then the Nikon J5.
I'm a fellow FF Sony shooter and have found myself very fond of the 35mm focal length for a walk-around lens with my kids. My 35mm is Sigma's F1.4 DG HSM, which comes in at 775 grams and is fairly large at 77mm x 120mm. I've found myself grabbing my Sony FE 50mm f1.8 due to its much more compact size and weight, so I'm strongly considering a slower, but more compact 35mm to grab when I'll be outdoors.
I don't know if this is the case for you, but I am actively choosing my gear around gear I'll be more likely to use - even if this comes comes with some trade-offs in terms of IQ or aperture speed. My gear blows a cellphone out of the water for my hyperactive kids, but as they say the best camera in the world is the one you're carrying.