A business card is about 0.23 mm thick and has a density of about 1g/cc
A credit card is about 0.8 mm thick and has a density of about 1.4 g/cc
Lithium has a density of 0.53 g/cc, but it is not a strong metal. It oxidizes easily and is not used as a pure metal.
Magnesium has a density of 1.738 g/cc, but pure magnesium is similar to aluminum in strength.
Beryllium has a density of 1.848 g/cc and is quite strong. Pure beryllium is rather brittle. A person might be able to break piece about the thickness of a credit card.
Aluminum has a density of 2.702 g/cc.
Titanium has a density of 4.5 g/cc. It is much stronger than aluminum. A piece that is much thinner than a piece of aluminum might be just as strong.
Many pure metals can be made much stronger by adding a small percentage of some other metal to make an alloy, but most such alloys are a little more dense than the pure metal. An aluminum-lithium alloy may be 10% less dense than aluminum and stronger than pure aluminum, but not a great deal stronger than the usual aluminum-copper alloys.
Beryllium seems to be the material that comes closest to your description.
Various Wikipedia articles and research done while collecting samples of the chemical elements. Credit and business card data from weighing and measuring.