Yes, besides Uttar Kanda sargas 73-76 of Valmiki Ramayana, Puranas also mention this incident of Killing of a shudra ascetic called Shambuka by Shree Rama. For instance in Padma Purana- Shristi Kanda- Section 1 - Chapter 35 - The Killing of Sudra Ascetic we find mention of this story.
Brief of the story - :
After killing of Ravana, once when Shree Rama was ruling his kingdom. A shudra ascetic named Shabuka was performing a severe penance desiring to have godhood with his body (i.e. I want to be a god, without casting off this body) in Shree Rama's country.
One day an old brāhmaṇa villager brought his dead son who died prematurely to the residence of Rama and said that "Rama’s wicked deed is evident that death has come to you"
Rama was surprised and asked Narada Muni who was present over there about the saying. Narada told Rama that In this kali yuga practice of penance will be found in the persons born as śūdras. A king, in whose kingdom or city, a wicked person does an unrighteous thing unfit to be done goes to hell and the king takes the fourth portion of his sin and that is why the Brahmana has said so. Narada then ordered Rama to find that person doing the wicked deed and try to stop him then and only then the boy's life span will be restored.
Rama immediately goes to Himalaya to find the deed and in the northern side of Himalaya big lake. In it a sage hanging upside down, practising very great penance. When asked the Shudra replied to Shree Rama
85-86a. O descendant of Kakutstha, know me to be a śūdra, Śambūka, by
name.” When he was thus talking, Rāma, the descendant of Kakutstha,
took out from the sheath, a spotless sword, and cut off his head.
86b-88a. When the Śūdra was killed, gods with Indra, led by Agni,
repeatedly praised Rāma (saying), ‘Well (done, well done)’; and a very
fragrant shower of flowers discharged by the gods from the sky,
scattered all around Rāghava.
Afterwards, when asked a boon by gods Shree Rama asked to restore the life of the Brahmana Boy.