Speedo makes (or at least used to make) a non-lycra swim suit. I have one, its the very plain style (classic one piece racer), but it has lasted forever. The lining which I think was lycra based gave out, but since it is dark blue so you can't see anything through it anyway. I used to swim in a very chlorinated pool (I worked in a hospital and the therapy pool was open for use by employees at night - they kept the water at about 92/93 degrees, it was like the worlds largest hot tub ) I did that several nights a week for at least 2 years and I still have the suit now, about 6 years later, and it looks exactly the same as the day I bought it.
I can't remember exactly, but I think that its made of polyester? It certainly is not nearly as stretchy as lycra, but it works just fine. I also used to use a really nice swim suit specific detergent to wash it. It was really good at removing chlorine - I couldn't smell anything at all on the suit after I washed it.