diff --git a/r/R/dplyr-mutate.R b/r/R/dplyr-mutate.R
index 051c5254e50..65421cd1e7f 100644
--- a/r/R/dplyr-mutate.R
+++ b/r/R/dplyr-mutate.R
@@ -38,11 +38,13 @@ mutate.arrow_dplyr_query <- function(.data,
   .data <- as_adq(.data)
 
   # Restrict the cases we support for now
-  if (length(dplyr::group_vars(.data)) > 0) {
+  has_aggregations <- any(unlist(lapply(exprs, all_funs)) %in% names(agg_funcs))
+  if (has_aggregations) {
+    # ARROW-13926
     # mutate() on a grouped dataset does calculations within groups
     # This doesn't matter on scalar ops (arithmetic etc.) but it does
     # for things with aggregations (e.g. subtracting the mean)
-    return(abandon_ship(call, .data, "mutate() on grouped data not supported in Arrow"))
+    return(abandon_ship(call, .data, "window functions not currently supported in Arrow"))
   }
 
   mask <- arrow_mask(.data)
diff --git a/r/R/util.R b/r/R/util.R
index 4811d5dbfbb..40b06fe959e 100644
--- a/r/R/util.R
+++ b/r/R/util.R
@@ -58,10 +58,29 @@ r_symbolic_constants <- c(
   "NA_integer_", "NA_real_", "NA_complex_", "NA_character_"
 )
 
+is_function <- function(expr, name) {
+  if (!is.call(expr)) {
+    return(FALSE)
+  } else {
+    if (deparse(expr[[1]]) == name) {
+      return(TRUE)
+    }
+    out <- lapply(expr, is_function, name)
+  }
+  any(map_lgl(out, isTRUE))
+}
+
 all_funs <- function(expr) {
-  # Don't use setdiff so that we preserve duplicates
-  out <- all.names(expr)
-  out[!(out %in% all.vars(expr))]
+  # It is not sufficient to simply do something like
+  #   setdiff(all.names, all.vars)
+  # here because that would fail to return the names of functions that
+  # share names with variables.
+  # To preserve duplicates, call `all.names()` not `all_names()` here.
+  if (is_quosure(expr)) {
+    expr <- quo_get_expr(expr)
+  }
+  names <- all.names(expr)
+  names[map_lgl(names, ~ is_function(expr, .))]
 }
 
 all_vars <- function(expr) {
diff --git a/r/tests/testthat/test-dataset.R b/r/tests/testthat/test-dataset.R
index 41265f0e638..14157545d61 100644
--- a/r/tests/testthat/test-dataset.R
+++ b/r/tests/testthat/test-dataset.R
@@ -772,7 +772,7 @@ test_that("mutate() features not yet implemented", {
     ds %>%
       group_by(int) %>%
       mutate(avg = mean(int)),
-    "mutate() on grouped data not supported in Arrow\nCall collect() first to pull data into R.",
+    "window functions not currently supported in Arrow\nCall collect() first to pull data into R.",
     fixed = TRUE
   )
 })
diff --git a/r/tests/testthat/test-dplyr-mutate.R b/r/tests/testthat/test-dplyr-mutate.R
index 44127839108..30fc12ccf17 100644
--- a/r/tests/testthat/test-dplyr-mutate.R
+++ b/r/tests/testthat/test-dplyr-mutate.R
@@ -340,14 +340,57 @@ test_that("dplyr::mutate's examples", {
   # The mutate operation may yield different results on grouped
   # tibbles because the expressions are computed within groups.
   # The following normalises `mass` by the global average:
-  # TODO: ARROW-11702
+  # TODO: ARROW-13926
   expect_dplyr_equal(
     input %>%
       select(name, mass, species) %>%
       mutate(mass_norm = mass / mean(mass, na.rm = TRUE)) %>%
       collect(),
     starwars,
-    warning = TRUE
+    warning = "window function"
+  )
+})
+
+test_that("Can mutate after group_by as long as there are no aggregations", {
+  expect_dplyr_equal(
+    input %>%
+      select(int, chr) %>%
+      group_by(chr) %>%
+      mutate(int = int + 6L) %>%
+      collect(),
+    tbl
+  )
+  expect_dplyr_equal(
+    input %>%
+      select(mean = int, chr) %>%
+      # rename `int` to `mean` and use `mean` in `mutate()` to test that
+      # `all_funs()` does not incorrectly identify it as an aggregate function
+      group_by(chr) %>%
+      mutate(mean = mean + 6L) %>%
+      collect(),
+    tbl
+  )
+  expect_warning(
+    tbl %>%
+      Table$create() %>%
+      select(int, chr) %>%
+      group_by(chr) %>%
+      mutate(avg_int = mean(int)) %>%
+      collect(),
+    "window functions not currently supported in Arrow; pulling data into R",
+    fixed = TRUE
+  )
+  expect_warning(
+    tbl %>%
+      Table$create() %>%
+      select(mean = int, chr) %>%
+      # rename `int` to `mean` and use `mean(mean)` in `mutate()` to test that
+      # `all_funs()` detects `mean()` despite the collision with a column name
+      group_by(chr) %>%
+      mutate(avg_int = mean(mean)) %>%
+      collect(),
+    "window functions not currently supported in Arrow; pulling data into R",
+    fixed = TRUE
   )
 })