According to Reddit, Russia's greatest cultural export of recent years has been hilarious dashcam videos. Inevitably, someone asks why dashcams are so prevalent, and someone else answers that insurance fraud is rampant in Russia. Is jumping in front of a car for the insurance money a post-Soviet development, or was there already a tradition established before the fall of communism? What would happen if a peasant was hit and injured by some low-level bureaucrat's Lada?
I assume you are asking about civil law in the sense of tort laws? In that case, it absolutely exists*: See Soviet Tort Law: The New Principles Annotated. The very first case is an auto accident, that ended with,
"The People's Court of Krasnopresnensk Region gave judgment for 576 rubles in favor of Autobase No. 12 against Tabulin for the damage inflicted.
*I mean that civil law exists, as oppose to speaking on insurance fraud in particular.
Interestingly enough, under the soviet system, it's up to the defendant to prove that he did not cause the injury (as oppose to the other way around under the American and most western systems),
"Under this article, the victim is not required to prove that the injury arose through the fault of the inflictor. The inflictor, provided he wants to be relieved of liability, must himself prove that he was not at fault in inflicting the injury. Thus, a person who has inflicted injury is presumed to have been at fault until he rebuts this presumption.
Cite: SoVETSKOYE GRAZHDANSKOYE PRAVO (SovIET CivI. LAW) 390 (Orlovskiy ed. 1961) via the above mentioned paper.
As to your specific example of a peasant being injured by a low-level bureaucrat in reality, I'm sure the story will be different depending on both time and place. I have some reasonable speculations, but I won't put them down in a top level post.
Edit: Added some details