It is fairly simple to search for 3x3 or 4x4 grids with this property, and I did so. There are only 5 Scrabble words (from SOWPODS) of length 3 that cannot be extended:
SWY FAP TWP VLY QIS
You can't make a grid out of this. A 2x2 grid won't work either as there are no two-letter words that can't be extended.
There are 250 nonextensible 4-letter words, and I found the only three 4x4 grids via exhaustive search (it's pretty feasible as long as you aggressively prune.)
THEY HIYA EYRY YAYS GJUS JIVY UVAE SYED TECS ELHI CHEZ SIZY
Unfortunately I think none of these are reachable (legal) Scrabble positions, particularly the first one with six Y's! So the question is still an open one. Perhaps this shape might be a fruitful avenue of exploration, if we can identify two-letter words that are extensible but not with a common letter:
.xx. xxxx xxxx .xx.
Otherwise the minimal stymied position might be considerably larger but take letter frequency into account.