Lead makes a good roof, but finding leaks in one (or any flat roof) can be a real bear as the leak below may be a long way from the leak above.
If the folded seams are (as I suspect from your description) standing up, except where walked on and flattened, then yes, the walked-on parts would be prime suspects for leakage. Sometimes the the seams are all folded down and soldered, but standing seams are not usually also soldered (it's a fair bit of fiddly torch work.)
Look very carefully for cracks right next to the flattened seams. If you don't see any cracks, run a torch over the flattened area (trying to anneal the metal before reworking it) and carefully stand the seam back up - look for cracks on the lower side as well. If there are cracks you (or a lead roof specialist you hire) can clean the area and solder them.
If there is a wall above the flat roof area, also look very carefully at the flashing where the wall joins the roof - likewise look very carefully around any penetrations in the roof.
Some folks will no doubt tell you to reroof, since lead is "old-fashioned" and therefore also "old" - depending on the shape the roof is in this might be the right thing to do, but both lead and terne-plate (lead coated steel) roofs can last for hundreds of years, so it is well worth trying to fix it first, IMHO.
Carefully pull the suspect nails (or all the nails) with a small pry bar or "cat's paw" and replace with new corrugated roofing nails like these:
![enter image description here](https://i.stack.imgur.com/ocKpT.jpg)
Or, if appearance is not important, smear a big glob of roof patch over the existing nails' heads.
Best Answer
No, this is not OK. Standing water will deteriorate any membrane and eventually cause the roofing and roof structure to deteriorate.
The wetting and drying process puts stresses on the roofing. The edge of a “puddle” has different coefficients of expansion and contraction. This different movement will reduce the life of the roofing.
Good roof design is intended to shed water. (There was a short time in the 1980’s that designed a “pond” on the roof to create an insulated barrier to the elements, but we soon learned that any imperfection will allow the entire “pond” drain through any pinhole.)
Steep roofs seem to perform better and lasts longer than flatter roofs. (Have you ever noticed that old barn roofs don’t seem to leak, even though you can look up and see daylight coming through the roof?)