Penrith is presumably actually a Welsh name
Sort of. There are two schools of thought, both broadly leading back to Welsh origins:
First is "Pen" (head/chief) "rid" or "ryhd" (ford) roughly translates as Head of the Ford. Both Pen and rid/rhyd come from Cumbric, which is a regional Brittonic language sharing much in common with Welsh.
Second is "Pen" (same as above - head/chief) "rhudd" (Welsh word for Crimson) relating to the red sandstone which is abundant locally. Could also relate to literally "Red Hill" after the hill on which Penrith Beacon (rip 86255) sits looking over the town. There is an area of Penrith called Redhills.
So there are Welsh/Cumbric origins and the town was probably once called Penrid, Penryhd or Penrhudd and became anglicised. The second explanation seems more likely as there isn't actually a river of any note running through the town, plus Penrhudd pronounced in Welsh isn't a million miles from Penrith (dd in Welsh = th in English, broadly)
I lived there for over a decade and my folks still do. I used to travel to Cornwall back in the 90s on the Cornish Scot every summer to spend time with a friend - Bodmin Parkway always tried selling her a ticket to Penryn rather than Penrith every time for the reverse journey.