Yes. A non-zero PSA level does not mean cancer [my Dr advised].
In my case [wrong side of 60] I have to take a piddle perhaps half the number of times the village clock chimes the hours through the night hours. Digital exam confirms enlarged prostate but nothing amiss [no roughness]. Thus my non-zero [but low] PSA level was seen as normal...in my case.
I have an annual PSA test to make sure that number does not change markedly. That seems to be the value of the PSA test: if there is a year-on-year increase it may portend more.
Just to reiterate the point
@zambezi makes above, a non-zero PSA is not necessarily a sign of a problem.
My case was very similar - elevated PSA scores resulted in a transperineal ultrasound biopsy. The results of the biopsy came back clear, and the diagnosis was that my PSA scores were high because I had an enlarged prostate.
The prostate is typically about the size of a walnut, but as we age the prostate can enlarge, and in some cases can grow to about the size of a lemon (though I wouldn't want a slice of it with my gin!). By the age of 60, half of all men will have an enlarged prostate, and by the age of 85 that proportion will have risen to around 95%. This is normal, so don't fear it.
Also having an enlarged prostate doesn't mean you will get prostate cancer, so if you are finding it difficult to pee, or that you are having to get up in the night to pee, don't automatically fear the worst. It may well be nothing worse than you now have a lemon pressing on your bladder rather than a walnut!
I have now had a couple of follow-up meetings with the consultant, who is happy with the progress, and I also have regular tests to monitor my PSA. It is typical that PSA scores go up, and go down, and what is being looked for is a persistent trend where the score is elevated or non-elevated.
Also be aware that certain activities can affect your PSA score, including vigorous exercise (particularly cycling) and ejaculation - and no, that doesn't only apply if you do both at the same time

. The recommendation is that you avoid these activities for at least two days prior to your PSA test (I should be so lucky.....).
Men are particularly poor at discussing their health, particularly when it concerns things "down there". However having a regular PSA test is probably one of the most single most important things any man can do in terms of protecting their long-term health, and it is encouraging to see it being so openly talked about here. Whatever you do, don't be afraid to discuss it - getting that PSA test can literally be a life-saver.