Is it permissible to keep an old metal coin—dating back to 37 AD—found lying on the road?

If the currency is gold or silver, or has value, and its minting date is before Islam, then it is a rikāz (buried treasure). One-fifth of its value or the object itself must be given as zakāt, and you may dispose of the remainder as you wish. However, if it has no value or is not gold or silver, then there is no objection to disposing of it as you wish. If it is highly probable that it belongs to a jewelry store owner or someone else, then it is a luqaṭa (found object) that must be announced for a year if it has significant value.