I was specifically commenting on manned flight but, yes, the Soviets had a rocketry lead that also allowed them to take the first pictures of the far side of the moon https://en.wikipedia.org/wiki/Luna_3 which is an interesting story of its own. (Basically they shot the photos on film and faxed them back to earth.) But they couldn't get the bigger rocket needed to send and return cosmonauts to and from the moon working.

Soviet Mars 3 was also the first probe to achieve soft landing on Mars. Like most of the Soviet Mars missions, it failed, but it failed only after landing (unlike most of the rest ...).