Can anyone explain why we say on a strictly cash basis (and not strict)? Is cash here an adjective? Is cash basis a noun phrase or compound noun?
Cash basis is a compound noun; it is frequently written as 'cash-basis'. The word preceding it ('strictly') is modifying a noun, so it is an adjective. The adjectival form of of this word is 'strict'. So, we should say:
... on a strict cash basis.
Strictly is an adverb, which can modify a verb, an adjective, or another adverb; but an adverb cannot modify a noun. If you really want to use 'strictly' instead of 'strict', then you need to use a sentence such as:
We strictly deal on a cash basis.
In this case, 'strictly' modifies the verb 'deal'. Ultimately, both sentences mean essentially the same thing, but you should not merge them by allowing 'strictly' to modify the compound noun.
However, it is not uncommon, although grammatically incorrect, to hear sentences such as, "We deal on a strictly cash basis', spoken in English speaking countries. One day, given enough time, this incorrect phrase may garner enough acceptance that it will be regarded as grammatically correct, and yet another grammar 'rule' will have been modified.