In books and movies I have heard of things costing half a penny I'm those days for example. How would they give you change if you wanted only one of whatever they were selling for (in this example) half a penny? Did they have a coin worth less than a penny?
Initially, English pennies from around the 10th Century onwards were minted with markings in them specifically designed to allow them to be split into halves and quarters. From the early 12th Century, actual halfpenny coinage started to appear in England. From the 13th Century onwards, silver farthing coinage (quarter of a penny) was minted, although the coinage wasn't entirely practical given how small it was. From the reign of James I onwards, the coin was struck in copper instead, which meant a bigger coin could be used.
In short, yes.
In long, Britain had two coins worth less than a penny:
the halfpenny (aka "ha'penny") was minted from medieval times all the way up until 1969 (and then again, post-decimalisation, from 1971-1984).
the farthing, worth a quarter of a penny, which also dates from the medieval era and was abandoned in 1961.
As a curiosity, there were also several sub-farthing coins which were only produced in certain British colonies (due to the lower cost of living there, a smaller coin was needed) - these were legal in the rest of the British Empire but unlikely to be found outside their "homes":
The half-farthing, produced in Ceylon (modern-day Sri Lanka) from 1828-1856, and demonetised in 1870.
The third-farthing, produced intermittently in Malta between 1827 and 1913.
The quarter-farthing, produced in Ceylon between 1839-1853.
As for the USA, it minted its own half-cent coin from 1793 until it was discontinued in 1857. The US also had the "mill", worth a tenth of a cent, which was mainly used for accounting and taxation purposes. Mill coins were never minted, but cheap mill tokens were sometimes issued by various states to help businesses manage their taxes.