The letter on your button is a P written in Old-English Script. Most often, that letter written in that "font" is a generic City (or County) Police button.
We time-date metal buttons by what the backmark says, and its "font," and other characteristics like the circle(s) of dots around the backmark, and the punctuation-marks in it. Your button's backmark says Superior * Quality * in what is called indented "plain block" lettering, with two indented stars or asterisks, all inside two circles of dots. That specific version of Quality-rating backmark began showing around the 1890s (plus or minus a few years), and was used through most of the 20th-Century. That being said, the "stars" or asterisks mostly went away by the mid-20th-Century. So I'd date your button to most likely being made between about 1900 and 1950.
