Surprisingly … no! Each verbal language has its own sign language.
Like natural languages, sign languages are developed locally, by people who interact with one another
In fact, British sign language and its American counterpart are so different that deaf people from both sides of the pond can’t understand each other.
Bonus Fact: although primarily developed for death people there are a few known cases where a population of hearing-people created spontaneously a sign language; But, even in these cases, languages were created alongside spoken languages, as a communication method for special situations where voice cannot be used (hunting, diving, etc).
Some of those rare cases where sign language formatted among hearers emerged among Aboriginal people in Australia. These languages evolved due to taboo rules of speaking in certain social situations such as mourning or initiation ceremonies.
For example, sign language was created among Aboriginal women, due to the social ban on widows speaking for months and sometimes even years after the death of their husband. Because widows often used to leave the community and live with only other widows or single women, this sign language became a common means of communication between those widows.