No, it is not. Moreover, if we take a broad historical perspective on the matter, we will see that chances for social mobility are much higher for people living in modern societies than for people who lived during the Middle Ages, for example.
Social mobility exists in every historical period. Once you have a society, this automatically means that different people occupy different positions within it, which usually means people can plausibly move between these positions as they move physically across space, as they age, as they undergo any life transition on the job, in the family, etc. So one cannot even create an imaginary society in one's head without also imagining social mobility.
Social mobility is a sociological term that denotes any transition from one social position to another. The first sociologist who defined this term was Pitirim Sorokin, who proposed that we should differentiate between vertical and horizontal social mobility. Vertical mobility can take place when there is unequal distribution of power and wealth among different people or groups; one has the possibility to move to a new position in the hierarchy and to become poorer or wealthier than s/he was earlier, to name one example. Horizontal social mobility means moving from one position to another that is more or less equal in terms of wealth and power distribution. For instance, one can change places of living, professions, jobs etc, without actually losing wealth, influence, power, or prestige.
We can say that over the course of human history in the Western world, social mobility has become more widespread in recent times. For instance, if you were born as a peasant in the Middle Ages, there was a very high chance that you would also die as a member of the peasantry. In the contemporary world, there are many more chances to move between different social positions. The fact that one person from a poor family can plausibly become wealthy during her/his adult life is not all that surprising now. Moreover, there is widespread horizontal mobility as people change jobs, and thus their positions in society, without climbing up the social ladder. However, social mobility varies greatly across different societies. It is the task of sociologists to to study the channels and the mechanisms of social mobility in different points in time and across different societies.