No, it's not even technically possible in the hardware, as some of the state like sprites is write-only. Well, it's read/write internally, but the rendering circuit is using the internal BRAM read port resources in such a way that the data is not currently exposed apart from inside the video signal. Even if it later becomes technically possible, the full state of a game is much more more distributed around the hardware than just copying the contents of main RAM, so it would be a challenge to write such a snapshotter. Even snapshotting the screen is non-trivial, as it's made up of stacked layers with transparency, stenciling, blend modes and dynamic copper effects, and the possible set of configurations those things can be used in (that a game can put them in) is enormous.
Generally, snapshotting only works well when you have a higher level of hardware than the game is running on itself. So Multiface can snapshot an original game because it has its own ROM and RAM (the Multiface in the Next works the same way for classic games). ResiDOS can snapshot and multitasks sets of complete 48K processes because it runs in the 128K extra RAM. PC emulators can snapshot everything the emulator can emulate, because they can access all the internals and have additional PC CPU/RAM resources. But all hardware on the Next is capable of being used in a Next-specific game, including the divMMC, the entire SD card, and (theoretically) the Multiface, so there is no external context that a snapshotter could run in.
The write-only thing is a slight problem on original machines too, for example border colour can't be snapshotted in a multiface snapshot, current 128K bank can't be snapshotted because the paging port is write-only, complete AY register state can't be snapshotted because the register select port is write only, etc. External hardware has to watch the Z80 bus with a state machine to guess some of that state. On the Next, the sprite port writes go out over the bus too, so external hardware or the Pi could theoretically track it, but it's orders of magnitude more complicated than tracking 128K paging. One thing we were planning on adding later is putting the current pixel RGB333 colour (the final output of the rendering circuit, over time) in a place that's readable by internal software or external hardware. That would avoid the need to have to decode the stacked layer configuration and contents to do a screenshot.