I have a commitment in the city that is keeping me here for a few days.
'Impegno' — stress the second syllable: 'im-PEG-no'. 'Trattiene' — 'trat-TIEN-e'.
Use as a polite, vague explanation for needing to stay longer — for work, meetings, or personal reasons you do not wish to elaborate on.
'Impegno' = commitment/engagement. 'Che mi trattiene' = 'that is keeping me (here)'. 'Trattenere' (to keep/detain) is a nuanced verb used for what holds you somewhere. 'Qualche giorno' = a few days.
Ho riunioni di lavoro che durano più del previsto.
I have work meetings that are lasting longer than expected.
States a professional reason for extension
Devo aspettare dei documenti importanti.
I have to wait for some important documents.
Administrative reason for staying longer
Resto per motivi di lavoro.
I am staying for work reasons.
Simple and discreet
Italian privacy culture (riservatezza) means that hotel staff will never probe into why you are staying. 'Impegni di lavoro' (work commitments) is accepted without question. Italian hotels serving business travellers are expert at flexible arrangements for extended stays.