CalfordMath

S=LAMBDA(p,LET(z,XMATCH(0,TOCOL(--p)),c,LAMBDA(c,x,IF(x>9,0,IF(NOT(OR(INDEX(p,INT((z-1)/9)+1,SEQUENCE(1,9))=x,INDEX(p,SEQUENCE(9),MOD(z-1,9)+1)=x,INDEX(p,SEQUENCE(3,1,INT(((z-1)/27))*3+1),SEQUENCE(1,3,FLOOR(MOD(z-1,9),3)+1))=x)),LET(r,S(IF(SEQUENCE(9,9,1)=z,x,p)),IF(AND(r),r,c(c,x+1))),c(c,x+1)))),IFERROR(c(c,1),p)));

Or a slightly different, even faster solution (332 characters):

S=LAMBDA(p,LET(n,SEQUENCE(9,9,0),z,XMATCH(0,TOCOL(--p))-1,g,LAMBDA(UNIQUE(VSTACK(SEQUENCE(9),TOCOL(INDEX(p,FLOOR(INT(z/9),3)+{1;2;3},FLOOR(MOD(z,9),3)+{1,2,3})),TOCOL(INDEX(p,INT(z/9)+1,)),INDEX(p,,MOD(z,9)+1)),,1)),c,LAMBDA(c,L,x,IF(x>ROWS(L),FALSE,LET(s,S(IF(n=z,INDEX(L,x),p)),IF(MIN(s)>0,s,c(c,L,x+1))))),IFERROR(c(c,g(),1),p)))

//=S(A1:I9) or
//=S({8,0,0,0,0,0,0,0,0;0,0,3,6,0,0,0,0,0;0,7,0,0,9,0,2,0,0;0,5,0,0,0,7,0,0,0;0,0,0,0,4,5,7,0,0;0,0,0,1,0,0,0,3,0;0,0,1,0,0,0,0,6,8;0,0,8,5,0,0,0,1,0;0,9,0,0,0,0,4,0,0})

//credits: 

CalfordMath

//ref: https://codereview.stackexchange.com/q/199771

//CalfordMath edit notes:
//I placed the "test" lambda inside of "solver" to acheive a single Lambda function (nested recursive). 
//I'm not sure this optimizes anything computationally, and perhaps it makes the code less easy to follow, 
//but I find it satisfying to have everything wrapped in a single function :)


solver = LAMBDA(grid,
    LET(

CalfordMath

Sudoku = LAMBDA(Puzzle,
    LET(
        Options, MAKEARRAY(27, 27, LAMBDA(r, c, MOD(c - 1, 3) + 1 + 3 * MOD(r - 1, 3))),
        numgrid, SEQUENCE(9, 9, 1),
        UpdatedPuzzle, LET(
            ApplyLogic, LAMBDA(ApplyLogic, Puzzle, Options,
                LET(
                    Candidates, MAKEARRAY(
                        27,
                        27,