diff --git a/src/utils/include/utils/math/bspline.hpp b/src/utils/include/utils/math/bspline.hpp
index 2a3ddf30e55..0dd677c606c 100644
--- a/src/utils/include/utils/math/bspline.hpp
+++ b/src/utils/include/utils/math/bspline.hpp
@@ -22,6 +22,11 @@
 #include "sqr.hpp"
 #include "utils/device_qualifier.hpp"
 
+#if defined(__CUDACC__)
+#include <cfloat>
+#else
+#include <limits>
+#endif
 #include <stdexcept>
 
 namespace Utils {
@@ -30,9 +35,17 @@ template <int order, typename T>
 DEVICE_QUALIFIER auto bspline(int i, T x) -> T
   requires((order > 0) and (order <= 7))
 {
+#if defined(__CUDACC__)
+  // LCOV_EXCL_START
+  [[maybe_unused]] auto constexpr tolerance = T(6) * (T)(FLT_EPSILON);
+  // LCOV_EXCL_STOP
+#else
+  [[maybe_unused]] auto constexpr tolerance =
+      T(6) * std::numeric_limits<T>::epsilon();
+#endif
   DEVICE_ASSERT(i < order);
-  DEVICE_ASSERT(x >= T(-0.5));
-  DEVICE_ASSERT(x <= T(0.5));
+  DEVICE_ASSERT(x >= T(-0.5) or (T(-0.5) - x) < tolerance);
+  DEVICE_ASSERT(x <= T(0.5) or (x - T(0.5)) < tolerance);
 
   switch (order) {
   case 1:
@@ -209,9 +222,17 @@ template <int order, typename T = double>
 DEVICE_QUALIFIER auto bspline_d(int i, T x) -> T
   requires((order > 0) and (order <= 7))
 {
+#if defined(__CUDACC__)
+  // LCOV_EXCL_START
+  [[maybe_unused]] auto constexpr tolerance = T(6) * (T)(FLT_EPSILON);
+  // LCOV_EXCL_STOP
+#else
+  [[maybe_unused]] auto constexpr tolerance =
+      T(6) * std::numeric_limits<T>::epsilon();
+#endif
   DEVICE_ASSERT(i < order);
-  DEVICE_ASSERT(x >= T(-0.5));
-  DEVICE_ASSERT(x <= T(0.5));
+  DEVICE_ASSERT(x >= T(-0.5) or (T(-0.5) - x) < tolerance);
+  DEVICE_ASSERT(x <= T(0.5) or (x - T(0.5)) < tolerance);
 
   switch (order - 1) {
   case 0:
diff --git a/testsuite/python/p3m_tuning_exceptions.py b/testsuite/python/p3m_tuning_exceptions.py
index bb84e2128ab..3a5e500c5f8 100644
--- a/testsuite/python/p3m_tuning_exceptions.py
+++ b/testsuite/python/p3m_tuning_exceptions.py
@@ -51,6 +51,7 @@ def get_valid_params(self, key, **custom_params):
     def setUp(self):
         self.system.box_l = [10., 10., 10.]
         self.system.periodicity = [True, True, True]
+        self.system.cell_system.skin = 0.1
 
     def tearDown(self):
         if espressomd.has_features(["ELECTROSTATICS"]):
@@ -65,7 +66,7 @@ def add_charged_particles(self):
         self.system.part.add(pos=[[0., 0., 0.], [0.5, 0.5, 0.5]], q=[-1., 1.])
 
     def add_magnetic_particles(self):
-        self.system.part.add(pos=[[0.01, 0.01, 0.01], [0.5, 0.5, 0.5]],
+        self.system.part.add(pos=[[0., 0., 0.], [0.5, 0.5, 0.5]],
                              dip=[(1., 0., 0.), (-1., 0., 0.)],
                              rotation=2 * [(True, True, True)])
 
@@ -622,6 +623,38 @@ def test_09_no_errors_dlc_dp3m_cpu_rescale_mesh(self):
             espressomd.magnetostatics.DLC,
             self.get_valid_params("DP3M", accuracy=0.1))
 
+    def check_extended_precision_edge_case(self, class_p3m, precisions):
+        # the following parameters lead to precision loss on extended precision
+        # architectures (80-bit floating-point arithmetic); particle positions
+        # in mesh units are lying outside the mesh domain by a tiny amount, but
+        # that shouldn't affect charge assignment (see PR #5135 for details)
+        length = 5.81825
+        system = self.system
+        system.box_l = 3 * [length]
+        system.time_step = 0.01
+        system.cell_system.skin = 0.  # skin=0 is mandatory to trigger the bug
+        system.part.add(pos=[[0., 0., 0.], [1., 1., 1.]], q=[-1., 1.])
+        for precision in precisions:
+            system.electrostatics.solver = class_p3m(
+                prefactor=2., accuracy=1e-2, mesh=3 * [10], cao=7, r_cut=1.5,
+                alpha=29.4842, tune=False, single_precision=precision)
+            system.integrator.run(0, recalc_forces=True)
+            f0 = np.copy(system.part.by_id(0).f)
+            f1 = np.copy(system.part.by_id(1).f)
+            np.testing.assert_allclose(f0, +0.34, rtol=0., atol=0.1)
+            np.testing.assert_allclose(f1, -0.34, rtol=0., atol=0.1)
+
+    @utx.skipIfMissingFeatures("P3M")
+    def test_09_no_errors_p3m_cpu_extended_precision_edge_case(self):
+        self.check_extended_precision_edge_case(
+            espressomd.electrostatics.P3M, precisions=[True, False])
+
+    @utx.skipIfMissingGPU()
+    @utx.skipIfMissingFeatures("P3M")
+    def test_09_no_errors_p3m_gpu_extended_precision_edge_case(self):
+        self.check_extended_precision_edge_case(
+            espressomd.electrostatics.P3MGPU, precisions=[True])
+
 
 if __name__ == "__main__":
     ut.main(failfast=True)