Was my mother a French citizen? I have read that children of French citizens automatically receive French citizenship, no matter where they are born.
That's correct. Your mother was born a French citizen and almost certainly remained one all her life. If that's the case, you would also have been born a French citizen but you may have lost the right to benefit from it later in life.
Someone told me that when she died the link was broken. Is this possible?
It's not that simple but it does start the clock on a rather complex rule that may deprieve second or third generation French citizens of their citizenship (article 30-3 of the civil code). You need to actively use French citizenship to maintain it while living abroad. If you do, it's possible for a family to keep it indefinitely, long after the death of the last French citizen born in France. Your mother obviously will not have been able to do that after her death so if she died more than 50 years ago, you are definitely not a French citizen anymore.
Otherwise, you can read Will it be possible to get French citizenship?, About losing French nationality by 'not-using', and French Citizenship and article 30-3 of the Civil Code to learn more about this rule. In your case, I think that the most important question is whether you are able to prove that your mother made use of her French citizenship “recently” (e.g. do you have proof that she requested a French passport or voted in a French election?).