-し is the classical/literary 終止形 of adjectives that doesn’t exist as a separate form anymore in modern Japanese, but it’s still commonly used with some words, especially 無し.
-たい can be used in this way. ソバを食べていたい means something like “I want to keep eating soba”. I don’t think it’s a likely thing for someone to say but it is possible.
But classical conjugations and leaving out the い in ている don’t go well together. Another thing is that 食ぶ (where 食べる comes from) is actually a humble expression for 食ふ (modern 食う). So when you see 食 together with -たし e.g. in a saying, it’ll likely be 食いたし and not 食べたし, for example in 河豚は食いたし命は惜しし.