If you mean old life as in “past lives” then no. Reincarnation is an illusion and the belief in reincarnation comes from ignorance.

If you mean old life as in “life before enlightenment” then sure. Memory is still there, it just becomes very factual, detached from emotion. Usually when we recall something we also to some extent relive the associated emotional experience. Which is why memories often spontaneously arise in the mind, triggered by the current emotional experience. When this is gone, memories arise in the mind much more rarely, up to the point where they pretty much don’t arise at all.

But it doesn’t mean they’re not there. If someone were to ask me “where did you live when you were 20 years old” it doesn’t mean I wouldn’t be able to answer. Same goes for frustrations, dreams and other forms of suffering.

A good analogy here is childhood memories. I bet you can recall something from your childhood, possibly with some effort, where you were upset, crying or frustrated about something you find absolutely trivial now. It will not give rise to any emotional experience in you, currently, it will be a factual recollection, or, at most, you will chuckle at your old baby self. So it’s kinda similar to that. On the other hand, if some childhood experience left a scar in you which you still carry, current events could spoteneously trigger a recollection (and reliving, in some form) of this experience, and it would be colored with emotion.