Yes. Give some dimensions for the calibre. looks like it has two sets of double firing pin marks so that suggests a Henry..but it seems bit too long case...
Yes an early rimfire cartridge case manufactured by Union Metallic Cartridge Co. hence U stamp. However I'm not familiar with the four circular firing pin marks on the case rim in comparison to say the .22 rimfire which leaves a rectangular mark on the case rim. Case possibly could be .32, or a .41 or .44. For definate caliber i.d. need to measure its total length using a caliper and diameter measure directly above case rim where it's not flattened. Measuring rim diameter can be helpful also. Than try Google early rimfire cartridge case dimensions.
Myself being interested in guns am interested in the 4 marks on case head and what type firearm made them.
I finally did a little Google research over coffee this morning and discovered the 1866 Winchester had 2 firing pins at 3 and 9 o'clock and fired the .44 rimfire. So thinking that cartridge did not fire the first time and was removed and rechambered slightly rotated and fired the second time.
Well, as the case seems to be fairly long and as the Winchester 1866 apparently did not have round firing pins. I am leaning towards your case being 10,4x38R swiss ...here we also call it just 'Vetterli'
Nots of those rifles were imported second hand in the USA and if memory serves newly made ammunition was available until ww2