Barnham has one village primary and a handful of secondaries reachable in the surrounding parishes. The in-village primary is Barnham Primary School. The nearest secondaries are Ormiston Six Villages Academy, St Philip Howard Catholic School and Felpham Community College. Chichester Free School functions as a sub-regional option that crosses the local-authority boundary. The current Ofsted picture for any of these is not carried in the data for this answer, so what follows describes the geography of choice rather than the verdict on teaching quality. Across the wider parish cluster, the primary network also includes Eastergate Church of England Primary School in the parish that merged with Barnham in 2019, plus Aldingbourne Primary School, Walberton and Binsted Church of England Primary School and Yapton Church of England Primary School at the edges of the catchment. The independent route is available too: Slindon College sits in the parish that gives it its name, and Westbourne House School lies further out toward Chichester.
The catchment caveat matters more here than the school list does. West Sussex allocates secondary places on a distance-and-feeder basis, which means a Barnham postcode inside Ormiston Six Villages Academy's notional catchment is the start of the conversation, not the end of it. Living within reach of Felpham Community College is not the same as a place at Felpham Community College, and the same is true at every named secondary in the area. Buyers who treat "good schools nearby" as the same thing as "my child will be at one of them" find out the difference at the offer-letter stage in March, by which point the house has been bought and the move is on. The honest version of the advice is to pick the postcode, identify the feeder primary, confirm the live catchment for the year your child enters, and treat the estate agent's catchment map as marketing copy rather than admissions policy. Barnham Primary School is a feeder for the academy cluster sitting around the village, which is part of why the village pricing holds up the way it does. Whether your child gets the secondary you wanted depends on where else they could have gone, who else applied, and what the council's distance calculation produces in the year of application.
The verdict is honest middle. There is a village primary, a small cluster of secondaries within reach, and an independent route if you want one. Workable, plannable, rarely the reason anyone moved to Barnham, and not the reason anyone leaves.