I'm sure you've heard about the EU's recent ban of EVA foam playmats related to the presence of formamide, a known carcinogenic chemical. From what we can tell, EVA foam – not standard EVA sheet material – is the source of concern because formamide is sometimes used to make the foam more pliable.
Please note that I said sometimes. It's possible to locate formamide-free EVA foam, although many manufacturers don't routinely check (or test) for it yet. We recently investigated a new product to stock in our store and the manufacturer was able to confirm with their factory that a small amount of formamide was present in their products. We were impressed with their decision to locate a new EVA foam source to ensure the safest product possible.
There is still some grey area surrounding this issue, because the EU hasn't determined exactly what levels are considered safe in children's products. It was noted though, that new mats have the highest risk and formamide residue decreases over time, so older products may contain little or no formamide.
EVA is recommended by CHEJ as a PVC-free substitute, as well as by Healthcare without Harm for use in neonatal intensive care units in drug delivery devices. The great thing about EVA is that it doesn't require the addition of plasticizers/softeners (like phthalates), stabilizers (like lead), or chlorine (which leads to the generation of dioxin, a highly persistent carcinogen).
Just be careful to research EVA foam products before buying and contact manufacturers to confirm that their products are formamide-free.