Is there a way to save a multi part print if one fails?

Mick is right that you can do this with OctoPrint and maybe other print managers, and 0scar is right that there's no way to get the printer to stop printing the no-longer-wanted part when printing without a print manager (since it doesn't know about objects). However there is a way to recover and keep the non-failed parts if you're willing to waste material printing the rest of the failed part. It involved having a recovery tool built in advance, though.

Basically, you need an adjustable-height platform you can temporarily glue to the bed where the failed part would have been, small enough not to interfere with other parts but large enough to cover at least most of the area the failed part should have been in. The surface of the platform needs to be something your material will adhere to, ideally the same material you're printing with.

With the print paused and the head parked to the side, glue the platform to the bed and adjust its height using the depth-measurement function of a caliper to match the current layer height. Alternatively, you could use a straight-edge resting on two adjacent printed parts to match the layer height, but that risks dislodging them.

After it's attached and adjusted properly, resume printing.

I have not tested this with tall parts, but I have successfully made such "platforms" with layers of blue tape for small parts, and see no reason the concept should not scale to arbitrary height as long as you can get sufficient adhesion and match the layer height sufficiently well. Designing a sufficiently adjustable and reusable one (with adjustment mechanism that doesn't interfere above the desired height) would be an interesting project.

If you use the OctoPrint print manager, you can exclude regions to be printed using the Exclude Region plugin. The description states that it can be used to rescue partially-failed prints:

The intent of this plugin is to provide a means to salvage multi-part prints where one (or more) of the parts has broken loose from the build plate or has otherwise become a worthless piece of failure. Instead of cancelling an entire job when only a portion is messed up, use this plugin to instruct OctoPrint to ignore any gcode commands executed within the area around the failure.

Other print managers may provide similar facilities.

I'm not sure if it will enable you to rescue your current print job.

No, once you sliced the 6 parts on the build plate in your slicer, the G-code is fixed and the printer will print as the sliced instructions. During printing it cannot skip the code of a part that failed along the way; there is no way to interfere with the printing other than stopping the print. For that reason, many people don't pack the build plate too full, the more parts, the higher the chance it fails. You could print the part one after each other. Packing the plate with multiple parts is usually not faster than printing one at a time (if it fails you have nothing, otherwise a single failed print).

Depending on the size of the parts you can also tell the printer to print each part on the plate one after another in a single job; note that the printhead dimensions limit this. If one fails you stop the job, but the already printed parts are saved. You can then commence a new print of cut the G-code and reprint the shortened file.