A Trial to Find the Safe Dose for BI 891065 Alone and in Combination With BI 754091 in Patients With Incurable Tumours or Tumours That Have Spread

Inclusion criteria: * Provision of signed and dated, written informed consent form (ICF) in accordance with International Conference of Harmonization-Good Clinical Practice (ICH-GCP) and local legislation prior to any trial-specific procedures, sampling, or analyses * Patients ≥18 years-of-age at the time of signature of the ICF * Male or female patients. Women of childbearing potential (WOCBP) and men able to father a child must be ready and able to use highly effective methods of birth control (that result in a low failure rate of less than 1% per year when used consistently and correctly)during trial participation and for at least 6 months after the last administration of trial medication. A list of contraception methods meeting these criteria is provided in the patient information. * Eastern Cooperative Oncology Group (ECOG) score: 0 or 1 * Life expectancy of at least 12 weeks after the start of the treatment according to the Investigator's judgement * For Parts A and B: Patients with a confirmed diagnosis of advanced, unresectable and/or metastatic solid tumours, who have failed standard treatment, or for whom no therapy of proven efficacy exists, or who are not amenable to standard therapies. Measurable lesions according to RECIST Version 1.1 must be present. Eligibility is limited to the following tumour subtypes in Part B: bladder, colon, breast, Non-small cell lung cancer (NSCLC), ovarian, pancreatic, renal, esophagogastric, sarcoma, prostate, and melanoma. * For Parts B and C: Patients must have measurable disease per RECIST v1.1, must have at least 1 tumour lesion amenable to biopsy, and must be willing to undergo a biopsy prior to first treatment and another biopsy while on therapy unless clinically contraindicated. * For Part C: Patients with metastatic NSCLC who developed disease progression (per RECIST v1.1) after the first scan (where SD, Partial Response (PR), or Complete Response (CR) was demonstrated at the first scan), and require new anti-cancer therapy after first line treatment with an anti programmed cell death protein 1 (PD-1)/anti programmed cell death ligand 1 (PD-L1) monoclonal antibody (mAb) (given either as single agent therapy or in combination with a platinum-based chemotherapy regimen). Exclusion criteria: * Major surgery (major according to the Investigator's and/or Medical Monitor's assessment) performed within 12 weeks prior to randomization or planned within 12 months after screening (e.g., hip replacement) * Presence of other active invasive cancers other than the one treated in this trial within 5 years prior to screening, except appropriately treated basal cell carcinoma of the skin, or in situ carcinoma of uterine cervix, or other local tumours considered cured by local treatment * Patients who must or wish to continue the intake of restricted medications or any drug considered likely to interfere with the safe conduct of the trial * Previous administration of BI 891065 or BI 754091 * Currently enrolled in another investigational device or drug trial, or less than 30 days since ending another investigational device or drug trial(s), or receiving other investigational treatments. * Patients who have been treated with any other anticancer drug within 4 weeks or within 5 half-life periods (whichever come earlier) prior to first administration of BI 891065. At least 7 days must have elapsed between the last dose of such agent and the first dose of study drug. * Persistent toxicity from previous treatments that has not resolved to ≤ Grade 1 (except for alopecia and Grade 2 neuropathy due to prior platinum-based therapy) * Active, known or suspected autoimmune disease except vitiligo or resolved asthma/atopy * Interstitial lung disease * Any of the following cardiac criteria: * Mean resting corrected QT interval (QTcF) \>470 msec * Any clinically important abnormalities (as assessed by the investigator) in rhythm, conduction, or morphology of resting Electrocardiograms (ECGs), e.g., complete left bundle branch block, third degree heart block * Any factors that increase the risk of QTc prolongation or risk of arrhythmic events such as heart failure, hypokalaemia, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years-of-age, or any concomitant medication with known or possible risk of QT interval prolongation * Patients with an ejection fraction (EF) \<55% or the lower limit of normal of the institutional standard will be excluded. Only in cases where the Investigator (or the treating physician or both) suspects cardiac disease with negative effect on the EF, will the EF be measured during screening using an appropriate method according to local standards to confirm eligibility (e.g., echocardiogram \[ECHO\], multi-gated acquisition scan \[MUGA\]). A historic measurement of EF no older than 6 months prior to first administration of study drug can be accepted provided that there is clinical evidence that the EF value has not worsened since this measurement in the opinion of the Investigator or of the treating physician or both. * Out of range laboratory values are defined as: * Alanine transaminase (ALT) and aspartate amino transferase (AST) \>3 times the Upper limit of normal (ULN) if no demonstrable liver metastases or \>5 times ULN in the presence of liver metastases * Total bilirubin \>1.5 times ULN, except for patients with Gilbert's syndrome who are excluded if total bilirubin \>3.0 times ULN or direct bilirubin \>1.5 times ULN * Human immunodeficiency virus (HIV) infection, acute or chronic viral hepatitis * Known hypersensitivity to the trial drugs or their excipients * Serious concomitant disease or medical condition affecting compliance with trial requirements or which are considered relevant for the evaluation of the efficacy or safety of the trial drug, such as cardiac, neurologic, psychiatric, infectious disease or active ulcers (gastro-intestinal tract, skin) or laboratory abnormality that may increase the risk associated with trial participation or trial drug administration, and in the judgment of the Investigator and/or Medical Monitor would make the patient inappropriate for entry into the trial. * Chronic alcohol or drug abuse or any condition that, in the Investigator's and/or Medical Monitor's opinion, makes them an unreliable trial patient or unlikely to complete the trial * Women who are pregnant, nursing, or who plan to become pregnant while in the trial and for at least 6 months after the last administration of trial medication. * Men who plan to father a child while in the trial and for at least 6 months after the last administration of trial medication. * Known presence of symptomatic central nervous system (CNS) metastases, unless asymptomatic and off corticosteroids and/or anti-convulsant therapy for at least 2 weeks prior start of treatment. Patients with asymptomatic CNS metastases may be enrolled following a 2-week washout period. * Patients receiving systemic treatment with any immunosuppressive medication within 1 week prior treatment start (steroids of max. 10 mg prednisolone equivalent per day are allowed, topical and inhaled steroids are not considered as immunosuppressive). * For Parts A and B: Patients with known epidermal growth factor receptor (EGFR), known anaplastic lymphoma kinase (ALK), or known ROS Proto-Oncogene 1 (ROS1) genomic tumour aberrations, unless disease has progressed following available EGFR or ALK targeted therapy (including osimertinib for EGFR T790M-mutated NSCLC) * Out of range lab values as defined: * Absolute neutrophil count (ANC) \<1.5 x 109/L (\<1500/mm3) * Platelet (PLT) count \<100 x 109/L * Haemoglobin \<90 g/L (\<9 g/dL) \-- Creatinine \>1.5 times ULN (patients may enter if creatinine is \>1.5 x ULN and estimated glomerular filtration rate (eGFR) \>30 mL/min/1.73 m2) (Chronic Kidney Disease Epidemiology \[CKD-EPI\] Collaboration equation); confirmation of eGRF is only required when creatinine is \>1.5 X ULN. * For Part C: Patients with EGFR, ALK, or (if known) ROS1 genomic tumor aberrations * For Part C: Patients with any CTLA-4 therapy * For Part C: One or more lines of anti-cancer therapy between previous anti-PD-1/anti-PDL1 mAb therapy and study entry.