The main purpose of this study is to learn more about the safety and side effects of the study drug known as LY3090106 in participants with Sjögren's Syndrome (SS). The study will also evaluate how much of the study drug gets into the blood stream and how long it takes the body to remove it.

Inclusion Criteria:

• Have a confirmed diagnosis of SS by the American-European Consensus Group criteria with active disease (at any level), as per judgment of the investigator (participants with another concurrent, stable connective tissue disease may be eligible for inclusion, with written approval from the sponsor).
• Are seropositive for auto-antibodies associated with SS (anti-SSA or anti-SSB) at screening, or documented within 6 months prior to screening.

Exclusion Criteria:

• Are currently enrolled in a clinical trial involving an investigational product or off-label use of a drug, are concurrently enrolled in any other type of medical research judged not to be scientifically or medically compatible with this study, or have received:

  • Any nonbiologic investigational product within 30 days or 5 half-lives (whichever is longer) of study baseline,
  • Any biologic investigational product within 3 months or 5 half-lives (whichever is longer) of study baseline, or any leukocyte depleting agent within 12 months of baseline,

• Have synthetic disease-modifying anti-rheumatic drug (DMARD) or immunosuppressive use as follows:

  • Any treatment with tofacitinib within 28 days prior to baseline or planned treatment with tofacitinib during the study,
  • Unstable prescribed dose of other synthetic DMARDs (eg, hydroxychloroquine, methotrexate, leflunomide, sulfasalazine, or gold salts) within 28 days prior to baseline or if the dose of drug is planned to be increased during the study. (Stable prescriptions are allowed.)
  • Cytotoxic or immunosuppressive drugs including but not limited to cyclophosphamide, mycophenolic acid, azathioprine, cyclosporine, sirolimus, or tacrolimus within 28 days prior to screening or planned treatment during the study.

• Have had treatment with biologic DMARDs as follows:

  • Etanercept, adalimumab, or anakinra <4 weeks before baseline or planned treatment during the study.
  • Infliximab, certolizumab pegol, golimumab, abatacept, or tocilizumab <8 weeks before baseline or planned treatment during the study.
  • Rituximab, belimumab or other leukocyte depleting agent <12 months before baseline or planned treatment during the study. Note: other biologic agents may be allowed after written approval from the sponsor.

• Have a prescribed dose >10 milligrams (mg)/day of oral prednisone (or equivalent) within 28 days before baseline, or plan to increase >10 mg/day during the study. (Stable prescriptions ≤10 mg/day are allowed.) Treatment with inhaled or parenteral corticosteroids within 28 days prior to baseline is prohibited. A single intra-articular corticosteroid injection is permitted within 28 days prior to baseline if no more than 40 mg triamcinolone (or equivalent) is administered. The treated joint should be excluded from any joint-specific evaluations during the study.

• Have an unstable prescribed dose of a cholinergic stimulant (eg, pilocarpine, cevimeline) within 28 days prior to baseline. (Stable prescriptions are allowed.)

• Have an unstable prescribed dose of cyclosporine eye drops within 28 days prior to baseline. (Stable prescriptions are allowed.)