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We compute the slice spectral sequence for the motivic stable homotopy groupsof $L$, a motivic analogue of the connective $K(1)$-local sphere over primefields of characteristic not two. Together with the analogous computation overalgebraically closed fields, this yields information about the motivic$K(1)$-local sphere over arbitrary base fields of characteristic not two. Tocompute the slice spectral sequence, we prove several results which may be ofindependent interest. We describe the $d_1$-differentials in the slice spectralsequence in terms of the motivic Steenrod operations over general base fields,building on analogous results of Ananyevskiy, R{\"o}ndigs, and {\O}stv{\ae}rfor the very effective cover of Hermitian K-theory. We also explicitly describethe coefficients of certain motivic Eilenberg--MacLane spectra and compute theslice spectral sequence for the very effective cover of Hermitian K-theory overprime fields.

The slice spectral sequence for a motivic analogue of the connective $K(1)$-local sphere