Is it considered showing off if a person informs others about a deed he did for the sake of Allah – after a period of time has passed since he did it – if his intention is not for them to imitate him?

If speaking about acts of obedience is intended to recount Allah's blessings and not for showing off or boasting, then there is no sin. However, if it is intended for praise or fame, then it is among the forbidden forms of showing off (sum'ah). It is advised to be sincere in acts of obedience and to avoid anything that contradicts it, such as ostentation (riya') or showing off (sum'ah).