Friction stir welding may be an option as it's a solid state welding process. As such 316 is having good weldability.

What method are you attempting to use to weld the foil? I would suggest micro-GTAW or micro-PAW, combined with pulsed current to minimize your heat input.

If you have the option to join the foil in overlap joint, try ultrasonic welding

 I would suggest micro-GTAW, micro-PAW or LBW for butt joints. If you use a different

joint like lap joint you may try the USW method. Just remember that you have to use the L type of stainless steel.   

I would suggest pulsed-laser welding for such a low thickness. As mentioned earlier micro-GTAW would also be a good choice.

For Butt Welding of 0.1 mm thickness of stainless steel use CO2 industrial laser welding and also try with plasma welding with low power .

I would suggest Electron Beam Welding (EBW) if it is Passible, Just remember that you must increase the welding speed.

We have successfully tried ultrasonic welding and pulsed Nd:YAG  laser.  When on USW try to keep the pneumatic clamp pressure low, otherwise overweld will occur. Also in Nd:YAG the beam dia has to be a bit more. This will prevent vaporization. I have not used CO2 for this  purpose, and it seems to me that material may burn at such high intensities.

I would recommend to use laser welding in heat conduction mode. The biggest problem here is usually not the welding process, but the clamping, as it has to be very precise.

I would recommend to use resistance welding (solid sate welding). Why because, it will generate less heat input.

When welding so thin foils, I recommend pulsed laser beam welding, due the low heat input. The use of solid state laser with small spot size (50 microns) for welding this type of stainless steel will bring good results. I agree with Mr. Barroi, that bigger issue will be the clamping of foils.

Best regards

Miroslav Sahul