Yes indeed.
I realised after I posted it that a generalised solution would have
been better so that it doesn't matter which column or which row the
data starts in.

Put the column letter of the first column of data in a cell somewhere
and name it "First" . In this example it should contain the letter A,
but if the data starts in another column just change the column letters

Then in V1 ( or anywhere else on row 1) put

=AVERAGE(INDIRECT(First&ROW()&":"&ADDRESS(ROW(U1), COLUMN(U1)-1)))

and copy it down